| L(s) = 1 | − 2.75·2-s + 3-s + 5.57·4-s + 1.28·5-s − 2.75·6-s − 1.78·7-s − 9.84·8-s + 9-s − 3.54·10-s − 0.289·11-s + 5.57·12-s + 13-s + 4.92·14-s + 1.28·15-s + 15.9·16-s + 5.29·17-s − 2.75·18-s − 5.86·19-s + 7.18·20-s − 1.78·21-s + 0.796·22-s − 1.13·23-s − 9.84·24-s − 3.34·25-s − 2.75·26-s + 27-s − 9.97·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.78·4-s + 0.575·5-s − 1.12·6-s − 0.676·7-s − 3.48·8-s + 0.333·9-s − 1.12·10-s − 0.0872·11-s + 1.61·12-s + 0.277·13-s + 1.31·14-s + 0.332·15-s + 3.98·16-s + 1.28·17-s − 0.648·18-s − 1.34·19-s + 1.60·20-s − 0.390·21-s + 0.169·22-s − 0.236·23-s − 2.01·24-s − 0.668·25-s − 0.539·26-s + 0.192·27-s − 1.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 11 | \( 1 + 0.289T + 11T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 23 | \( 1 + 1.13T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 + 8.62T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 + 0.831T + 43T^{2} \) |
| 47 | \( 1 + 0.285T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 9.83T + 71T^{2} \) |
| 73 | \( 1 + 1.48T + 73T^{2} \) |
| 79 | \( 1 - 0.519T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 9.14T + 89T^{2} \) |
| 97 | \( 1 - 1.52T + 97T^{2} \) |
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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
−8.188232953526005223072691848821, −7.63623868975858713769841036809, −6.87418903152484743346813530183, −6.19111854460168434836988539119, −5.56301974723899345119458369254, −3.83842110237089406705127983516, −2.97638282254371332788964119349, −2.14399148688508417206746529257, −1.36609032533618079771079572089, 0,
1.36609032533618079771079572089, 2.14399148688508417206746529257, 2.97638282254371332788964119349, 3.83842110237089406705127983516, 5.56301974723899345119458369254, 6.19111854460168434836988539119, 6.87418903152484743346813530183, 7.63623868975858713769841036809, 8.188232953526005223072691848821
