| L(s) = 1 | + 4·3-s + 8·7-s + 8·9-s + 8·11-s − 16-s + 8·17-s + 32·21-s − 16·23-s + 12·27-s + 12·31-s + 32·33-s − 16·37-s − 20·41-s − 8·47-s − 4·48-s + 32·49-s + 32·51-s − 16·59-s + 40·61-s + 64·63-s − 8·67-s − 64·69-s − 36·71-s + 64·77-s − 24·79-s + 23·81-s + 16·83-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.02·7-s + 8/3·9-s + 2.41·11-s − 1/4·16-s + 1.94·17-s + 6.98·21-s − 3.33·23-s + 2.30·27-s + 2.15·31-s + 5.57·33-s − 2.63·37-s − 3.12·41-s − 1.16·47-s − 0.577·48-s + 32/7·49-s + 4.48·51-s − 2.08·59-s + 5.12·61-s + 8.06·63-s − 0.977·67-s − 7.70·69-s − 4.27·71-s + 7.29·77-s − 2.70·79-s + 23/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.822365694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.822365694\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1654 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 540 T^{3} + 3854 T^{4} - 540 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1500 T^{3} + 10094 T^{4} + 1500 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 3922 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 6706 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 312 T^{3} + 2258 T^{4} + 312 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 36 T + 648 T^{2} + 8244 T^{3} + 79918 T^{4} + 8244 p T^{5} + 648 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^3$ | \( 1 - 8158 T^{4} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 186 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1584 T^{3} + 19346 T^{4} - 1584 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 11720 T^{3} + 133282 T^{4} + 11720 p T^{5} + 800 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.305932900390498528576978680219, −8.103645492056759134544126717953, −7.74600758109760758181571918743, −7.67473713429974482825475670958, −7.55619685371716340327695172926, −6.87506529497890108397211558061, −6.82784461445592411577122814801, −6.64576173622429365717023510167, −6.33281589595593858360264244791, −5.65756487521199693724810349655, −5.64078576697681484101894722440, −5.49584521567246337362634603620, −4.80685068546408908400018945439, −4.67078249787413561754978093867, −4.49058948311609807700447243127, −4.21033963135710380772805765840, −3.69872469437457057688333060901, −3.65953106061035151762540686789, −3.34970963925518150962030017727, −3.00778718609753474396255848504, −2.43829213377367628490749607329, −1.98063964093039899510980987278, −1.71589603282482398482603345638, −1.53250035823986984547594801172, −1.27518416345169607560151125301,
1.27518416345169607560151125301, 1.53250035823986984547594801172, 1.71589603282482398482603345638, 1.98063964093039899510980987278, 2.43829213377367628490749607329, 3.00778718609753474396255848504, 3.34970963925518150962030017727, 3.65953106061035151762540686789, 3.69872469437457057688333060901, 4.21033963135710380772805765840, 4.49058948311609807700447243127, 4.67078249787413561754978093867, 4.80685068546408908400018945439, 5.49584521567246337362634603620, 5.64078576697681484101894722440, 5.65756487521199693724810349655, 6.33281589595593858360264244791, 6.64576173622429365717023510167, 6.82784461445592411577122814801, 6.87506529497890108397211558061, 7.55619685371716340327695172926, 7.67473713429974482825475670958, 7.74600758109760758181571918743, 8.103645492056759134544126717953, 8.305932900390498528576978680219
