| L(s) = 1 | + 12·5-s + 8·9-s − 16·13-s + 24·25-s − 52·29-s + 196·37-s − 80·41-s + 96·45-s + 116·49-s − 144·53-s − 188·61-s − 192·65-s + 72·73-s − 46·81-s + 360·89-s + 64·97-s + 32·101-s − 20·109-s − 168·113-s − 128·117-s + 168·121-s − 348·125-s + ⋯ |
| L(s) = 1 | + 12/5·5-s + 8/9·9-s − 1.23·13-s + 0.959·25-s − 1.79·29-s + 5.29·37-s − 1.95·41-s + 2.13·45-s + 2.36·49-s − 2.71·53-s − 3.08·61-s − 2.95·65-s + 0.986·73-s − 0.567·81-s + 4.04·89-s + 0.659·97-s + 0.316·101-s − 0.183·109-s − 1.48·113-s − 1.09·117-s + 1.38·121-s − 2.78·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.385376050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.385376050\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 8 T^{2} + 110 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 6 T + 42 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 116 T^{2} + 7078 T^{4} - 116 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 168 T^{2} + 33006 T^{4} - 168 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 22 p T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 1124 T^{2} + 559078 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 164 T^{2} + 353158 T^{4} - 164 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 26 T + 1834 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 3380 T^{2} + 4649830 T^{4} - 3380 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 98 T + 5122 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 40 T + 2062 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 268 T^{2} + 6280006 T^{4} + 268 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 6276 T^{2} + 18492294 T^{4} - 6276 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 72 T + 3582 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 11124 T^{2} + 54490566 T^{4} - 11124 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 94 T + 4738 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 2612 T^{2} + 30465286 T^{4} - 2612 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 732 T^{2} + 43641606 T^{4} + 732 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 15524 T^{2} + 133966534 T^{4} - 15524 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 15156 T^{2} + 134794374 T^{4} - 15156 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 180 T + 22854 T^{2} - 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 32 T - 578 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.525896746322363201764117632654, −7.940030218112785150432515001044, −7.79788954296663645699560403406, −7.72124472473386846137170277617, −7.38922255122665929735584539608, −7.38250885271551341019823124789, −6.55096268689320774927346413175, −6.50738389509877523703988151333, −6.40656827060117175541887817752, −5.90945330790316154873614451818, −5.85711861240828335116114490021, −5.58611683126985654951338920449, −5.30025976775306466002713892623, −4.77183462468678694542450305989, −4.63199414628382318367313379568, −4.40197300840195694415728561966, −4.01992788133143296741630801109, −3.48758071652924378166576775383, −3.19671065179674943628356651431, −2.55815471707344003678999543947, −2.41571756484014103028563064135, −2.00574567941062320623237125318, −1.72256136683073846333847024883, −1.27685643868276377869788851535, −0.48751669314162118547837842698,
0.48751669314162118547837842698, 1.27685643868276377869788851535, 1.72256136683073846333847024883, 2.00574567941062320623237125318, 2.41571756484014103028563064135, 2.55815471707344003678999543947, 3.19671065179674943628356651431, 3.48758071652924378166576775383, 4.01992788133143296741630801109, 4.40197300840195694415728561966, 4.63199414628382318367313379568, 4.77183462468678694542450305989, 5.30025976775306466002713892623, 5.58611683126985654951338920449, 5.85711861240828335116114490021, 5.90945330790316154873614451818, 6.40656827060117175541887817752, 6.50738389509877523703988151333, 6.55096268689320774927346413175, 7.38250885271551341019823124789, 7.38922255122665929735584539608, 7.72124472473386846137170277617, 7.79788954296663645699560403406, 7.940030218112785150432515001044, 8.525896746322363201764117632654
