| L(s) = 1 | + 3-s + 2.49·5-s − 3.10·7-s + 9-s + 0.206·11-s − 4.17·13-s + 2.49·15-s + 4.17·19-s − 3.10·21-s + 4.44·23-s + 1.20·25-s + 27-s − 9.74·29-s − 2.80·31-s + 0.206·33-s − 7.72·35-s + 11.5·37-s − 4.17·39-s + 3.90·41-s − 0.206·43-s + 2.49·45-s + 10.0·47-s + 2.61·49-s + 5.23·53-s + 0.515·55-s + 4.17·57-s + 7.67·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.11·5-s − 1.17·7-s + 0.333·9-s + 0.0624·11-s − 1.15·13-s + 0.643·15-s + 0.958·19-s − 0.676·21-s + 0.927·23-s + 0.241·25-s + 0.192·27-s − 1.80·29-s − 0.504·31-s + 0.0360·33-s − 1.30·35-s + 1.90·37-s − 0.668·39-s + 0.609·41-s − 0.0315·43-s + 0.371·45-s + 1.46·47-s + 0.372·49-s + 0.718·53-s + 0.0695·55-s + 0.553·57-s + 0.999·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.620437157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.620437157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 - 0.206T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 - 4.44T + 23T^{2} \) |
| 29 | \( 1 + 9.74T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 + 0.206T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 8.38T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 - 2.33T + 73T^{2} \) |
| 79 | \( 1 - 8.40T + 79T^{2} \) |
| 83 | \( 1 - 5.27T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76682048396631659283523532758, −7.27528157933672159769717681834, −6.63754658597472381964799695271, −5.73203462237852376830170956820, −5.37295267271981958659164489241, −4.26144713543283897028113909406, −3.41961171445296548996059120980, −2.63476646798419851178610870059, −2.07971549763628612254481365082, −0.78734561800865320502684924701,
0.78734561800865320502684924701, 2.07971549763628612254481365082, 2.63476646798419851178610870059, 3.41961171445296548996059120980, 4.26144713543283897028113909406, 5.37295267271981958659164489241, 5.73203462237852376830170956820, 6.63754658597472381964799695271, 7.27528157933672159769717681834, 7.76682048396631659283523532758
