| L(s) = 1 | + 2.53·2-s − 3-s + 4.41·4-s + 0.933·5-s − 2.53·6-s − 7-s + 6.10·8-s + 9-s + 2.36·10-s + 2.77·11-s − 4.41·12-s − 5.52·13-s − 2.53·14-s − 0.933·15-s + 6.64·16-s − 2.51·17-s + 2.53·18-s − 2.48·19-s + 4.11·20-s + 21-s + 7.01·22-s − 6.14·23-s − 6.10·24-s − 4.12·25-s − 13.9·26-s − 27-s − 4.41·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 0.577·3-s + 2.20·4-s + 0.417·5-s − 1.03·6-s − 0.377·7-s + 2.15·8-s + 0.333·9-s + 0.747·10-s + 0.835·11-s − 1.27·12-s − 1.53·13-s − 0.676·14-s − 0.241·15-s + 1.66·16-s − 0.609·17-s + 0.596·18-s − 0.569·19-s + 0.921·20-s + 0.218·21-s + 1.49·22-s − 1.28·23-s − 1.24·24-s − 0.825·25-s − 2.74·26-s − 0.192·27-s − 0.833·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 - 0.933T + 5T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 + 4.88T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 + 3.83T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 + 1.28T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 - 4.29T + 53T^{2} \) |
| 59 | \( 1 + 0.733T + 59T^{2} \) |
| 61 | \( 1 - 1.81T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 9.63T + 71T^{2} \) |
| 73 | \( 1 + 3.84T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.51T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.07244775489462608166358187807, −6.57136221820659006561413220497, −5.86147610813532766938563745965, −5.47881496597644087447646529340, −4.60153546784024957238321153600, −4.09440256433340207538551870737, −3.39579429374033957140505758997, −2.24333482985371359359291796852, −1.87971632020916476232314623547, 0,
1.87971632020916476232314623547, 2.24333482985371359359291796852, 3.39579429374033957140505758997, 4.09440256433340207538551870737, 4.60153546784024957238321153600, 5.47881496597644087447646529340, 5.86147610813532766938563745965, 6.57136221820659006561413220497, 7.07244775489462608166358187807
