| L(s) = 1 | + 2.22·2-s + 2.93·4-s − 2.84·5-s − 1.80·7-s + 2.08·8-s − 6.32·10-s − 1.40·11-s + 5.09·13-s − 4.00·14-s − 1.24·16-s + 0.501·17-s + 7.81·19-s − 8.35·20-s − 3.13·22-s + 23-s + 3.09·25-s + 11.3·26-s − 5.29·28-s − 29-s − 7.79·31-s − 6.93·32-s + 1.11·34-s + 5.12·35-s + 6.33·37-s + 17.3·38-s − 5.92·40-s − 4.39·41-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.46·4-s − 1.27·5-s − 0.680·7-s + 0.736·8-s − 1.99·10-s − 0.424·11-s + 1.41·13-s − 1.06·14-s − 0.311·16-s + 0.121·17-s + 1.79·19-s − 1.86·20-s − 0.667·22-s + 0.208·23-s + 0.618·25-s + 2.22·26-s − 0.999·28-s − 0.185·29-s − 1.39·31-s − 1.22·32-s + 0.191·34-s + 0.866·35-s + 1.04·37-s + 2.81·38-s − 0.936·40-s − 0.686·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 - 0.501T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 37 | \( 1 - 6.33T + 37T^{2} \) |
| 41 | \( 1 + 4.39T + 41T^{2} \) |
| 43 | \( 1 + 8.99T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 7.18T + 61T^{2} \) |
| 67 | \( 1 - 9.87T + 67T^{2} \) |
| 71 | \( 1 - 0.384T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 + 4.12T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 5.51T + 89T^{2} \) |
| 97 | \( 1 - 9.04T + 97T^{2} \) |
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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
−7.66069108573466614176150681227, −6.74503077526930901384989528651, −6.25501013487178644758931104695, −5.36263348467815469278483930043, −4.84793415670438353753149651555, −3.85660851149208882895589617192, −3.39417628301106814099766840106, −3.03479789508223267904670080163, −1.51924151777585864888877593013, 0,
1.51924151777585864888877593013, 3.03479789508223267904670080163, 3.39417628301106814099766840106, 3.85660851149208882895589617192, 4.84793415670438353753149651555, 5.36263348467815469278483930043, 6.25501013487178644758931104695, 6.74503077526930901384989528651, 7.66069108573466614176150681227
