| L(s) = 1 | + 1.29·2-s + 3-s − 0.326·4-s + 5-s + 1.29·6-s − 0.504·7-s − 3.00·8-s + 9-s + 1.29·10-s − 3.12·11-s − 0.326·12-s − 4.12·13-s − 0.653·14-s + 15-s − 3.24·16-s + 0.706·17-s + 1.29·18-s − 2.41·19-s − 0.326·20-s − 0.504·21-s − 4.04·22-s − 3.00·24-s + 25-s − 5.33·26-s + 27-s + 0.164·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.914·2-s + 0.577·3-s − 0.163·4-s + 0.447·5-s + 0.528·6-s − 0.190·7-s − 1.06·8-s + 0.333·9-s + 0.409·10-s − 0.942·11-s − 0.0942·12-s − 1.14·13-s − 0.174·14-s + 0.258·15-s − 0.810·16-s + 0.171·17-s + 0.304·18-s − 0.554·19-s − 0.0730·20-s − 0.110·21-s − 0.861·22-s − 0.614·24-s + 0.200·25-s − 1.04·26-s + 0.192·27-s + 0.0311·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.945597400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.945597400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 7 | \( 1 + 0.504T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 - 0.706T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 0.560T + 31T^{2} \) |
| 37 | \( 1 - 9.72T + 37T^{2} \) |
| 41 | \( 1 - 7.87T + 41T^{2} \) |
| 43 | \( 1 - 9.78T + 43T^{2} \) |
| 47 | \( 1 - 7.66T + 47T^{2} \) |
| 53 | \( 1 - 9.88T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 2.50T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−7.74333442927837924268321566884, −7.20932250824769709922714782876, −6.15380831245445034600715051618, −5.75882422981931574824114049601, −4.82825829297540203898258536584, −4.44953996487983866749364753759, −3.54980066139962442519435716878, −2.55972083481290000278890270842, −2.41347467217687672837005580764, −0.69069095857084611535629192151,
0.69069095857084611535629192151, 2.41347467217687672837005580764, 2.55972083481290000278890270842, 3.54980066139962442519435716878, 4.44953996487983866749364753759, 4.82825829297540203898258536584, 5.75882422981931574824114049601, 6.15380831245445034600715051618, 7.20932250824769709922714782876, 7.74333442927837924268321566884
