| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s + 10·5-s + 24·6-s + 32·8-s + 27·9-s + 40·10-s + 50·11-s + 72·12-s + 100·13-s + 60·15-s + 80·16-s − 34·17-s + 108·18-s + 46·19-s + 120·20-s + 200·22-s + 126·23-s + 192·24-s + 75·25-s + 400·26-s + 108·27-s − 150·29-s + 240·30-s + 74·31-s + 192·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s + 1.37·11-s + 1.73·12-s + 2.13·13-s + 1.03·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s + 0.555·19-s + 1.34·20-s + 1.93·22-s + 1.14·23-s + 1.63·24-s + 3/5·25-s + 3.01·26-s + 0.769·27-s − 0.960·29-s + 1.46·30-s + 0.428·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(31.46074912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.46074912\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 - 50 T + 272 p T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 100 T + 6599 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 p T + 2740 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 46 T + 9527 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 126 T + 4408 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 150 T + 51748 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 74 T + 18471 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 176 T + 85155 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 52612 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 132 T + 148915 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 316 T + 227890 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 236 T + 282178 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 58 T - 296696 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 792 T + 606058 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 868 T + 740027 T^{2} + 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 386 T + 568696 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1220 T + 1083759 T^{2} - 1220 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 54 T + 415687 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 182 T + 54160 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 418 T + 1297564 T^{2} - 418 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 392 T + 1821282 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.094229025092102865961461605684, −9.041926338339617182678379675329, −8.470337946048167294647210111701, −8.398143476071925759473278268451, −7.41530762574035779223272217139, −7.35802873873328593631699592520, −6.71871749001100604507280158906, −6.42407120330293695981959501465, −6.08132815375001335769661928125, −5.71076487475928800848307531825, −5.02760485157906296580681600639, −4.77903478435272098168643400618, −3.98158970850887214605229644002, −3.85161703202682648812568484545, −3.20598308135652779887244848463, −3.19573514048918458342672773065, −2.10454562102327424559059022274, −2.04798191150843489528214511793, −1.12802414054931319449910072722, −1.08266430857376849043219485380,
1.08266430857376849043219485380, 1.12802414054931319449910072722, 2.04798191150843489528214511793, 2.10454562102327424559059022274, 3.19573514048918458342672773065, 3.20598308135652779887244848463, 3.85161703202682648812568484545, 3.98158970850887214605229644002, 4.77903478435272098168643400618, 5.02760485157906296580681600639, 5.71076487475928800848307531825, 6.08132815375001335769661928125, 6.42407120330293695981959501465, 6.71871749001100604507280158906, 7.35802873873328593631699592520, 7.41530762574035779223272217139, 8.398143476071925759473278268451, 8.470337946048167294647210111701, 9.041926338339617182678379675329, 9.094229025092102865961461605684
