| L(s) = 1 | − 1.58·2-s + 1.81·3-s + 0.518·4-s − 2.87·6-s + 0.0644·7-s + 2.35·8-s + 0.281·9-s − 3.11·11-s + 0.939·12-s + 1.08·13-s − 0.102·14-s − 4.76·16-s + 7.86·17-s − 0.446·18-s − 4.43·19-s + 0.116·21-s + 4.94·22-s + 1.31·23-s + 4.25·24-s − 1.72·26-s − 4.92·27-s + 0.0334·28-s + 9.34·29-s − 7.65·31-s + 2.86·32-s − 5.64·33-s − 12.4·34-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 1.04·3-s + 0.259·4-s − 1.17·6-s + 0.0243·7-s + 0.831·8-s + 0.0937·9-s − 0.939·11-s + 0.271·12-s + 0.302·13-s − 0.0273·14-s − 1.19·16-s + 1.90·17-s − 0.105·18-s − 1.01·19-s + 0.0254·21-s + 1.05·22-s + 0.274·23-s + 0.869·24-s − 0.338·26-s − 0.947·27-s + 0.00632·28-s + 1.73·29-s − 1.37·31-s + 0.506·32-s − 0.982·33-s − 2.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 3 | \( 1 - 1.81T + 3T^{2} \) |
| 7 | \( 1 - 0.0644T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 - 7.86T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 - 9.34T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 + 5.72T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 + 7.30T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 + 0.891T + 61T^{2} \) |
| 67 | \( 1 + 0.205T + 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 + 1.46T + 73T^{2} \) |
| 79 | \( 1 - 6.60T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 - 8.77T + 97T^{2} \) |
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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
−7.990689328334599949784617689121, −7.43573003940593553374713611949, −6.61981544283178913013452882338, −5.51534403587480831322026434133, −4.88572105051458982602029843079, −3.75925741685217848370532841880, −3.10587766823623292856629901853, −2.18214727235316509802508919572, −1.29825123572419170560920879402, 0,
1.29825123572419170560920879402, 2.18214727235316509802508919572, 3.10587766823623292856629901853, 3.75925741685217848370532841880, 4.88572105051458982602029843079, 5.51534403587480831322026434133, 6.61981544283178913013452882338, 7.43573003940593553374713611949, 7.990689328334599949784617689121
