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 + 32·11-s − 72·12-s + 12·13-s + 60·15-s + 80·16-s − 4·17-s + 108·18-s − 64·19-s − 120·20-s + 128·22-s + 32·23-s − 192·24-s + 75·25-s + 48·26-s − 108·27-s − 84·29-s + 240·30-s + 72·31-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 + 0.877·11-s − 1.73·12-s + 0.256·13-s + 1.03·15-s + 5/4·16-s − 0.0570·17-s + 1.41·18-s − 0.772·19-s − 1.34·20-s + 1.24·22-s + 0.290·23-s − 1.63·24-s + 3/5·25-s + 0.362·26-s − 0.769·27-s − 0.537·29-s + 1.46·30-s + 0.417·31-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
\(L(2)\) |
\(\approx\) |
\(5.809830013\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.809830013\) |
\(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - 32 T + 1222 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 12 T - 2354 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 8134 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 64 T - 522 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 32 T + 22894 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 84 T + 35278 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 72 T + 54094 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 60 T + 100510 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 92 T + 31414 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 184 T + 165782 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 208 T + 3746 p T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 300 T + 304990 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 568 T + 464278 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 516 T + 478126 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 392 T + 597542 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1448 T + 1212862 T^{2} + 1448 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 756 T + 751318 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 416 T + 1022558 T^{2} - 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1752 T + 1666726 T^{2} - 1752 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 956 T + 1529878 T^{2} - 956 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 684 T + 795814 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.308312095760455415504478218186, −8.960519624678742715666579753504, −8.421794099673240846342823758873, −8.016757496737506977353519632283, −7.39469216216241113928577805961, −7.26604751281023337146076915473, −6.70848156952837448965288614498, −6.28810172147534701338148745620, −6.16589304691016442256181253240, −5.61717487407936594178042817261, −4.97456142356607328467228436010, −4.85834516147405295198146694045, −4.27292215979894942725320433921, −3.94371970749990348771511845463, −3.52375161356166878632604775377, −3.08217761711216436483311500403, −2.16428515679126103716300865794, −1.83818106319281272416747959592, −0.862386267906064029963969317947, −0.58819448459192855493325233948,
0.58819448459192855493325233948, 0.862386267906064029963969317947, 1.83818106319281272416747959592, 2.16428515679126103716300865794, 3.08217761711216436483311500403, 3.52375161356166878632604775377, 3.94371970749990348771511845463, 4.27292215979894942725320433921, 4.85834516147405295198146694045, 4.97456142356607328467228436010, 5.61717487407936594178042817261, 6.16589304691016442256181253240, 6.28810172147534701338148745620, 6.70848156952837448965288614498, 7.26604751281023337146076915473, 7.39469216216241113928577805961, 8.016757496737506977353519632283, 8.421794099673240846342823758873, 8.960519624678742715666579753504, 9.308312095760455415504478218186