| L(s) = 1 | + 1.11·2-s + 0.936·3-s − 0.767·4-s + 1.03·6-s + 2.02·7-s − 3.07·8-s − 2.12·9-s + 0.335·11-s − 0.718·12-s + 1.60·13-s + 2.24·14-s − 1.87·16-s + 0.0940·17-s − 2.35·18-s − 8.15·19-s + 1.89·21-s + 0.372·22-s + 8.72·23-s − 2.87·24-s + 1.78·26-s − 4.79·27-s − 1.55·28-s − 2.89·29-s − 0.472·31-s + 4.06·32-s + 0.314·33-s + 0.104·34-s + ⋯ |
| L(s) = 1 | + 0.785·2-s + 0.540·3-s − 0.383·4-s + 0.424·6-s + 0.765·7-s − 1.08·8-s − 0.707·9-s + 0.101·11-s − 0.207·12-s + 0.445·13-s + 0.600·14-s − 0.468·16-s + 0.0228·17-s − 0.555·18-s − 1.86·19-s + 0.413·21-s + 0.0794·22-s + 1.81·23-s − 0.587·24-s + 0.349·26-s − 0.923·27-s − 0.293·28-s − 0.537·29-s − 0.0848·31-s + 0.718·32-s + 0.0547·33-s + 0.0179·34-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 | \( 1 - 1.11T + 2T^{2} \) |
| 3 | \( 1 - 0.936T + 3T^{2} \) |
| 7 | \( 1 - 2.02T + 7T^{2} \) |
| 11 | \( 1 - 0.335T + 11T^{2} \) |
| 13 | \( 1 - 1.60T + 13T^{2} \) |
| 17 | \( 1 - 0.0940T + 17T^{2} \) |
| 19 | \( 1 + 8.15T + 19T^{2} \) |
| 23 | \( 1 - 8.72T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 0.470T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 8.33T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 5.02T + 53T^{2} \) |
| 59 | \( 1 + 5.99T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 + 9.82T + 71T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 + 7.18T + 79T^{2} \) |
| 83 | \( 1 + 0.911T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.88830810811316240674961058749, −6.91298964453243861938980180961, −6.10977825334478935773454057165, −5.48653475860691909709500883223, −4.69975226164137798826093729918, −4.14885023904261026520773919464, −3.26898235913810892932997321846, −2.61868318243937275834758699383, −1.51999878706564871292204756487, 0,
1.51999878706564871292204756487, 2.61868318243937275834758699383, 3.26898235913810892932997321846, 4.14885023904261026520773919464, 4.69975226164137798826093729918, 5.48653475860691909709500883223, 6.10977825334478935773454057165, 6.91298964453243861938980180961, 7.88830810811316240674961058749
