| L(s) = 1 | + 1.35·2-s − 0.151·4-s − 3.36·5-s + 2.75·7-s − 2.92·8-s − 4.57·10-s + 0.0931·11-s + 3.43·13-s + 3.73·14-s − 3.67·16-s + 3.92·17-s − 0.647·19-s + 0.510·20-s + 0.126·22-s − 9.37·23-s + 6.30·25-s + 4.66·26-s − 0.417·28-s + 4.54·29-s − 3.37·31-s + 0.857·32-s + 5.33·34-s − 9.24·35-s + 0.224·37-s − 0.880·38-s + 9.83·40-s − 10.6·41-s + ⋯ |
| L(s) = 1 | + 0.961·2-s − 0.0759·4-s − 1.50·5-s + 1.03·7-s − 1.03·8-s − 1.44·10-s + 0.0280·11-s + 0.951·13-s + 0.999·14-s − 0.918·16-s + 0.951·17-s − 0.148·19-s + 0.114·20-s + 0.0269·22-s − 1.95·23-s + 1.26·25-s + 0.915·26-s − 0.0789·28-s + 0.844·29-s − 0.606·31-s + 0.151·32-s + 0.914·34-s − 1.56·35-s + 0.0368·37-s − 0.142·38-s + 1.55·40-s − 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 0.0931T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 19 | \( 1 + 0.647T + 19T^{2} \) |
| 23 | \( 1 + 9.37T + 23T^{2} \) |
| 29 | \( 1 - 4.54T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 0.224T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 6.39T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 - 0.241T + 61T^{2} \) |
| 67 | \( 1 + 9.03T + 67T^{2} \) |
| 71 | \( 1 + 1.03T + 71T^{2} \) |
| 73 | \( 1 - 9.39T + 73T^{2} \) |
| 79 | \( 1 - 0.898T + 79T^{2} \) |
| 83 | \( 1 + 6.81T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 6.44T + 97T^{2} \) |
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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
−8.190056646901398658744733802359, −7.47975107258774479759510436990, −6.43586408447824574528377168503, −5.66268584448775662083911584216, −4.91358368084344856227991566800, −4.06861416867424151622770225179, −3.84259777277216599626343856496, −2.87037305373965012563964665316, −1.42297467931087420861062062776, 0,
1.42297467931087420861062062776, 2.87037305373965012563964665316, 3.84259777277216599626343856496, 4.06861416867424151622770225179, 4.91358368084344856227991566800, 5.66268584448775662083911584216, 6.43586408447824574528377168503, 7.47975107258774479759510436990, 8.190056646901398658744733802359
