| L(s) = 1 | + 3.08·3-s − 5-s − 7-s + 6.52·9-s + 11-s − 5.79·13-s − 3.08·15-s + 1.08·17-s − 4.17·19-s − 3.08·21-s − 5.52·23-s + 25-s + 10.8·27-s + 0.703·29-s − 7.90·31-s + 3.08·33-s + 35-s − 9.52·37-s − 17.8·39-s − 3.73·41-s − 5.35·43-s − 6.52·45-s + 6.38·47-s + 49-s + 3.35·51-s + 5.52·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.78·3-s − 0.447·5-s − 0.377·7-s + 2.17·9-s + 0.301·11-s − 1.60·13-s − 0.796·15-s + 0.263·17-s − 0.957·19-s − 0.673·21-s − 1.15·23-s + 0.200·25-s + 2.09·27-s + 0.130·29-s − 1.42·31-s + 0.537·33-s + 0.169·35-s − 1.56·37-s − 2.86·39-s − 0.583·41-s − 0.816·43-s − 0.972·45-s + 0.930·47-s + 0.142·49-s + 0.469·51-s + 0.758·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 - 0.703T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 9.52T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 - 2.26T + 59T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 - 0.648T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 5.08T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.70241886227315257964025434249, −7.25024204469461652066245509269, −6.64767932910854979327305742561, −5.45112627668086001542700526909, −4.49802071056887313347853146583, −3.84734958306081155830684982323, −3.23309353073981094448261691526, −2.35734415112063024572934306784, −1.76371102152688934910494483886, 0,
1.76371102152688934910494483886, 2.35734415112063024572934306784, 3.23309353073981094448261691526, 3.84734958306081155830684982323, 4.49802071056887313347853146583, 5.45112627668086001542700526909, 6.64767932910854979327305742561, 7.25024204469461652066245509269, 7.70241886227315257964025434249
