L(s) = 1 | − 2-s − 2.57·3-s + 4-s + 4.04·5-s + 2.57·6-s + 7-s − 8-s + 3.64·9-s − 4.04·10-s − 6.22·11-s − 2.57·12-s − 3.30·13-s − 14-s − 10.4·15-s + 16-s + 3.51·17-s − 3.64·18-s − 3.74·19-s + 4.04·20-s − 2.57·21-s + 6.22·22-s − 8.78·23-s + 2.57·24-s + 11.3·25-s + 3.30·26-s − 1.64·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s + 1.80·5-s + 1.05·6-s + 0.377·7-s − 0.353·8-s + 1.21·9-s − 1.27·10-s − 1.87·11-s − 0.743·12-s − 0.917·13-s − 0.267·14-s − 2.68·15-s + 0.250·16-s + 0.851·17-s − 0.857·18-s − 0.859·19-s + 0.903·20-s − 0.562·21-s + 1.32·22-s − 1.83·23-s + 0.525·24-s + 2.26·25-s + 0.648·26-s − 0.317·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 2.57T + 3T^{2} \) |
| 5 | \( 1 - 4.04T + 5T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 8.78T + 23T^{2} \) |
| 29 | \( 1 + 0.0905T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 - 9.62T + 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 + 6.26T + 47T^{2} \) |
| 53 | \( 1 + 0.187T + 53T^{2} \) |
| 59 | \( 1 + 4.44T + 59T^{2} \) |
| 61 | \( 1 - 0.849T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 0.441T + 73T^{2} \) |
| 79 | \( 1 - 4.90T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 4.98T + 97T^{2} \) |
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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
−7.994878551879504357391769183262, −6.73773785666964772391317005388, −6.26509574174685393806803299255, −5.57057735767808295034570883944, −5.25086717171969778208106437365, −4.46975057164325625057247666788, −2.62961214811974343868185895965, −2.24987827034228277995406843739, −1.13325695776476882981539761385, 0,
1.13325695776476882981539761385, 2.24987827034228277995406843739, 2.62961214811974343868185895965, 4.46975057164325625057247666788, 5.25086717171969778208106437365, 5.57057735767808295034570883944, 6.26509574174685393806803299255, 6.73773785666964772391317005388, 7.994878551879504357391769183262