| L(s) = 1 | + 2-s + 2.14·3-s + 4-s − 0.309·5-s + 2.14·6-s + 7-s + 8-s + 1.60·9-s − 0.309·10-s − 5.74·11-s + 2.14·12-s + 0.589·13-s + 14-s − 0.664·15-s + 16-s − 6.54·17-s + 1.60·18-s − 4.09·19-s − 0.309·20-s + 2.14·21-s − 5.74·22-s − 4.05·23-s + 2.14·24-s − 4.90·25-s + 0.589·26-s − 2.99·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.23·3-s + 0.5·4-s − 0.138·5-s + 0.875·6-s + 0.377·7-s + 0.353·8-s + 0.533·9-s − 0.0979·10-s − 1.73·11-s + 0.619·12-s + 0.163·13-s + 0.267·14-s − 0.171·15-s + 0.250·16-s − 1.58·17-s + 0.377·18-s − 0.938·19-s − 0.0692·20-s + 0.468·21-s − 1.22·22-s − 0.844·23-s + 0.437·24-s − 0.980·25-s + 0.115·26-s − 0.577·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 5 | \( 1 + 0.309T + 5T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 - 0.589T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 + 0.169T + 29T^{2} \) |
| 31 | \( 1 + 7.45T + 31T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 - 6.71T + 47T^{2} \) |
| 53 | \( 1 + 9.35T + 53T^{2} \) |
| 59 | \( 1 - 3.72T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 - 8.30T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 + 5.88T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 8.50T + 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 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
−7.78506254916729134851567498534, −7.24285363583363540872321400867, −6.15658152634534169960284455840, −5.57678829259276181452912794300, −4.55476838759832667070764190390, −4.09294401132257832268830122652, −3.18102939782136968781309844134, −2.25110466894097691789700267898, −2.08991263894910158707015253006, 0,
2.08991263894910158707015253006, 2.25110466894097691789700267898, 3.18102939782136968781309844134, 4.09294401132257832268830122652, 4.55476838759832667070764190390, 5.57678829259276181452912794300, 6.15658152634534169960284455840, 7.24285363583363540872321400867, 7.78506254916729134851567498534
