| L(s) = 1 | + 1.21·2-s + 0.688·3-s − 0.525·4-s + 5-s + 0.836·6-s − 3.06·8-s − 2.52·9-s + 1.21·10-s − 11-s − 0.361·12-s + 3.73·13-s + 0.688·15-s − 2.67·16-s − 0.0666·17-s − 3.06·18-s − 6.42·19-s − 0.525·20-s − 1.21·22-s − 1.09·23-s − 2.11·24-s + 25-s + 4.54·26-s − 3.80·27-s − 7.80·29-s + 0.836·30-s + 5.59·31-s + 2.88·32-s + ⋯ |
| L(s) = 1 | + 0.858·2-s + 0.397·3-s − 0.262·4-s + 0.447·5-s + 0.341·6-s − 1.08·8-s − 0.841·9-s + 0.384·10-s − 0.301·11-s − 0.104·12-s + 1.03·13-s + 0.177·15-s − 0.668·16-s − 0.0161·17-s − 0.722·18-s − 1.47·19-s − 0.117·20-s − 0.258·22-s − 0.228·23-s − 0.431·24-s + 0.200·25-s + 0.890·26-s − 0.732·27-s − 1.44·29-s + 0.152·30-s + 1.00·31-s + 0.510·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 3 | \( 1 - 0.688T + 3T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 0.0666T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.595484453760818248915368379877, −7.899177443398826138501594175849, −6.54918024141490205210006027191, −6.05291780102710828405609301276, −5.34554628954152084259100760190, −4.46856700557285369464874371256, −3.61506901020544943838692872074, −2.89952424658914189787766920623, −1.84495277599261576460629129936, 0,
1.84495277599261576460629129936, 2.89952424658914189787766920623, 3.61506901020544943838692872074, 4.46856700557285369464874371256, 5.34554628954152084259100760190, 6.05291780102710828405609301276, 6.54918024141490205210006027191, 7.899177443398826138501594175849, 8.595484453760818248915368379877
