L(s) = 1 | + 4·5-s + 4·7-s − 16·11-s + 32·13-s − 40·17-s + 56·23-s + 16·25-s − 16·31-s + 16·35-s − 64·37-s − 56·41-s + 8·43-s + 128·47-s + 8·49-s − 56·53-s − 64·55-s − 200·61-s + 128·65-s + 200·67-s − 272·71-s + 76·73-s − 64·77-s − 9·81-s + 16·83-s − 160·85-s + 128·91-s − 20·97-s + ⋯ |
L(s) = 1 | + 4/5·5-s + 4/7·7-s − 1.45·11-s + 2.46·13-s − 2.35·17-s + 2.43·23-s + 0.639·25-s − 0.516·31-s + 0.457·35-s − 1.72·37-s − 1.36·41-s + 8/43·43-s + 2.72·47-s + 8/49·49-s − 1.05·53-s − 1.16·55-s − 3.27·61-s + 1.96·65-s + 2.98·67-s − 3.83·71-s + 1.04·73-s − 0.831·77-s − 1/9·81-s + 0.192·83-s − 1.88·85-s + 1.40·91-s − 0.206·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.296372805\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.296372805\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T - 4 p^{2} T^{3} + p^{4} T^{4} \) |
good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 156 T^{3} + 2942 T^{4} - 156 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 204 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 9120 T^{3} + 148994 T^{4} - 9120 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 15240 T^{3} + 281858 T^{4} + 15240 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 50904 T^{3} + 1508162 T^{4} - 50904 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 1722 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 64 T + 2048 T^{2} + 58176 T^{3} + 1440962 T^{4} + 58176 p^{2} T^{5} + 2048 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 28 T + 3342 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 5256 T^{3} - 557566 T^{4} - 5256 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 128 T + 8192 T^{2} - 506496 T^{3} + 28260194 T^{4} - 506496 p^{2} T^{5} + 8192 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} + 155064 T^{3} + 15333122 T^{4} + 155064 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 100 T + 7998 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 1888200 T^{3} + 153742658 T^{4} - 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 68 T + p^{2} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T + 2888 T^{2} + 65436 T^{3} - 36833458 T^{4} + 65436 p^{2} T^{5} + 2888 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 11882 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 101328 T^{3} + 79904642 T^{4} - 101328 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 173820 T^{3} + 150551438 T^{4} + 173820 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.86704090572562902401007087701, −6.80605635375447995600414701778, −6.61546980019509243437712428500, −6.17658396366123208818128895370, −6.14397470348236770145803749560, −5.84184254182392411419360411078, −5.64448852254817428679515064569, −5.31547067562833720662951148501, −5.06888386081432548987594602697, −4.99464212160568233787117158962, −4.75127434454207242969339712386, −4.49021250771653457460045267612, −4.06176879061453619766494921648, −3.86376818647487780596245319473, −3.70413043769854087468342690817, −3.31625102162055895034362973060, −2.89303568623781285310913328360, −2.74916042269715485301059886576, −2.53687835867811447663255922298, −2.23091509852329864483018942935, −1.52575840362480868456605172604, −1.52344328202321822850418770658, −1.41326740569199722506815108460, −0.73423915449271714187503884576, −0.16620950371646642288376835278,
0.16620950371646642288376835278, 0.73423915449271714187503884576, 1.41326740569199722506815108460, 1.52344328202321822850418770658, 1.52575840362480868456605172604, 2.23091509852329864483018942935, 2.53687835867811447663255922298, 2.74916042269715485301059886576, 2.89303568623781285310913328360, 3.31625102162055895034362973060, 3.70413043769854087468342690817, 3.86376818647487780596245319473, 4.06176879061453619766494921648, 4.49021250771653457460045267612, 4.75127434454207242969339712386, 4.99464212160568233787117158962, 5.06888386081432548987594602697, 5.31547067562833720662951148501, 5.64448852254817428679515064569, 5.84184254182392411419360411078, 6.14397470348236770145803749560, 6.17658396366123208818128895370, 6.61546980019509243437712428500, 6.80605635375447995600414701778, 6.86704090572562902401007087701