| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 3.23·11-s + 12-s + 3.23·13-s + 14-s + 15-s + 16-s − 2.47·17-s − 18-s − 4.47·19-s + 20-s − 21-s + 3.23·22-s + 23-s − 24-s + 25-s − 3.23·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.975·11-s + 0.288·12-s + 0.897·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s − 0.599·17-s − 0.235·18-s − 1.02·19-s + 0.223·20-s − 0.218·21-s + 0.689·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s − 0.634·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.640910175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640910175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 0.472T + 83T^{2} \) |
| 89 | \( 1 - 7.23T + 89T^{2} \) |
| 97 | \( 1 - 5.70T + 97T^{2} \) |
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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
−8.434155870949413074614177549538, −7.73085484875391877877317537249, −6.86884969193298084274851577905, −6.31034354799074678245610747258, −5.50674131385642535565527901560, −4.49763089799424932740494120511, −3.55450675398024210235827571674, −2.62284177555990265232042361189, −2.02683967629609381229260916704, −0.74883801951199862721704159242,
0.74883801951199862721704159242, 2.02683967629609381229260916704, 2.62284177555990265232042361189, 3.55450675398024210235827571674, 4.49763089799424932740494120511, 5.50674131385642535565527901560, 6.31034354799074678245610747258, 6.86884969193298084274851577905, 7.73085484875391877877317537249, 8.434155870949413074614177549538
