| L(s) = 1 | − 1.65·2-s + 2.37·3-s + 0.726·4-s − 3.92·6-s + 0.377·7-s + 2.10·8-s + 2.65·9-s − 1.37·11-s + 1.72·12-s + 2.82·13-s − 0.622·14-s − 4.92·16-s − 6.37·17-s − 4.37·18-s + 0.896·21-s + 2.27·22-s − 6.19·23-s + 4.99·24-s − 4.65·26-s − 0.829·27-s + 0.273·28-s + 3.37·29-s − 2.48·31-s + 3.92·32-s − 3.27·33-s + 10.5·34-s + 1.92·36-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 1.37·3-s + 0.363·4-s − 1.60·6-s + 0.142·7-s + 0.743·8-s + 0.883·9-s − 0.415·11-s + 0.498·12-s + 0.782·13-s − 0.166·14-s − 1.23·16-s − 1.54·17-s − 1.03·18-s + 0.195·21-s + 0.484·22-s − 1.29·23-s + 1.02·24-s − 0.913·26-s − 0.159·27-s + 0.0517·28-s + 0.627·29-s − 0.445·31-s + 0.693·32-s − 0.569·33-s + 1.80·34-s + 0.320·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 7 | \( 1 - 0.377T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 3.22T + 67T^{2} \) |
| 71 | \( 1 - 2.30T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 + 18.2T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−7.76430112430987670765885086236, −7.05626848082661800642428140007, −6.31298672222544791922792500939, −5.32101669702446436867846073361, −4.19688507188781376876827279335, −3.95198201199874126923237961211, −2.66880575040094026851613959156, −2.19202879052080751066400050014, −1.28805309762109242467223706384, 0,
1.28805309762109242467223706384, 2.19202879052080751066400050014, 2.66880575040094026851613959156, 3.95198201199874126923237961211, 4.19688507188781376876827279335, 5.32101669702446436867846073361, 6.31298672222544791922792500939, 7.05626848082661800642428140007, 7.76430112430987670765885086236
