| L(s) = 1 | + 4·3-s + 10·9-s + 8·11-s − 8·17-s − 8·19-s + 20·27-s + 32·33-s + 8·41-s + 24·43-s − 4·49-s − 32·51-s − 32·57-s + 48·67-s − 24·73-s + 35·81-s − 8·83-s − 40·89-s − 16·97-s + 80·99-s + 32·107-s − 8·113-s + 36·121-s + 32·123-s + 127-s + 96·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 2.41·11-s − 1.94·17-s − 1.83·19-s + 3.84·27-s + 5.57·33-s + 1.24·41-s + 3.65·43-s − 4/7·49-s − 4.48·51-s − 4.23·57-s + 5.86·67-s − 2.80·73-s + 35/9·81-s − 0.878·83-s − 4.23·89-s − 1.62·97-s + 8.04·99-s + 3.09·107-s − 0.752·113-s + 3.27·121-s + 2.88·123-s + 0.0887·127-s + 8.45·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.06813385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.06813385\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 22 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 2962 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 114 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 + 48 T^{2} + 3314 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 128 T^{2} + 8658 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 + 188 T^{2} + 17638 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 260 T^{2} + 28662 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.29756102179574667615210429387, −5.87386447699301822609986216269, −5.69246641488014104716292947345, −5.67995418562427987600867370592, −5.66946054496783088609642736698, −4.84381195408923648268449314870, −4.77763655895573287619933819692, −4.58947500203371519272038907622, −4.29451632163815742387043741432, −4.23312658390905247393341042548, −3.99948753103343983426563002597, −3.97275163145043905964329358111, −3.83674516681170750904735836126, −3.40468603912596007378707164479, −3.13630051986624178903541571163, −2.96282212973609135692457227542, −2.64777224013084646246781026595, −2.43396122023279445965090836283, −2.14656754034713115355310412126, −2.11107528956419902792577627896, −1.86632529778747746872437235824, −1.36675926702826513573526540961, −1.27215684959203948349697707972, −0.76530466628735849695128019441, −0.46635298350904294158324033656,
0.46635298350904294158324033656, 0.76530466628735849695128019441, 1.27215684959203948349697707972, 1.36675926702826513573526540961, 1.86632529778747746872437235824, 2.11107528956419902792577627896, 2.14656754034713115355310412126, 2.43396122023279445965090836283, 2.64777224013084646246781026595, 2.96282212973609135692457227542, 3.13630051986624178903541571163, 3.40468603912596007378707164479, 3.83674516681170750904735836126, 3.97275163145043905964329358111, 3.99948753103343983426563002597, 4.23312658390905247393341042548, 4.29451632163815742387043741432, 4.58947500203371519272038907622, 4.77763655895573287619933819692, 4.84381195408923648268449314870, 5.66946054496783088609642736698, 5.67995418562427987600867370592, 5.69246641488014104716292947345, 5.87386447699301822609986216269, 6.29756102179574667615210429387
