| L(s) = 1 | + i·2-s − i·3-s − 4-s + (2 + i)5-s + 6-s − 2i·7-s − i·8-s − 9-s + (−1 + 2i)10-s − 2·11-s + i·12-s − 2i·13-s + 2·14-s + (1 − 2i)15-s + 16-s − 6i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.894 + 0.447i)5-s + 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s − 0.603·11-s + 0.288i·12-s − 0.554i·13-s + 0.534·14-s + (0.258 − 0.516i)15-s + 0.250·16-s − 1.45i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58030 - 0.373058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58030 - 0.373058i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2 - i)T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.834232487658171719532765214120, −9.258641200621032622349906959493, −8.034241419903732867579777149116, −7.22897813146381006534951537156, −6.90816978743354370729078188135, −5.51534312916078272336124634205, −5.27388207243613997790328100551, −3.57913206698963702191270578467, −2.48263301015210684269712312384, −0.827578532815388943043566540273,
1.50113647328993692059699899371, 2.60679097366355926524178899677, 3.68981082289944354077731147925, 4.95701155419698269502046728540, 5.47584513291779601327362567906, 6.43261538440133354631313820935, 7.975424352582576624268774178038, 8.727133041099525573745914304987, 9.520603337532747126115513738503, 9.992703882407293044710947233183
