| L(s) = 1 | + 3.73·2-s − 3·3-s + 5.94·4-s − 11.2·6-s − 11.3·7-s − 7.67·8-s + 9·9-s − 55.6·11-s − 17.8·12-s + 20.9·13-s − 42.5·14-s − 76.2·16-s + 29.8·17-s + 33.6·18-s + 25.7·19-s + 34.1·21-s − 207.·22-s − 32.9·23-s + 23.0·24-s + 78.0·26-s − 27·27-s − 67.7·28-s − 29·29-s − 281.·31-s − 223.·32-s + 166.·33-s + 111.·34-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.743·4-s − 0.762·6-s − 0.615·7-s − 0.339·8-s + 0.333·9-s − 1.52·11-s − 0.429·12-s + 0.446·13-s − 0.812·14-s − 1.19·16-s + 0.425·17-s + 0.440·18-s + 0.311·19-s + 0.355·21-s − 2.01·22-s − 0.298·23-s + 0.195·24-s + 0.588·26-s − 0.192·27-s − 0.457·28-s − 0.185·29-s − 1.63·31-s − 1.23·32-s + 0.880·33-s + 0.561·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.014660508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014660508\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 3.73T + 8T^{2} \) |
| 7 | \( 1 + 11.3T + 343T^{2} \) |
| 11 | \( 1 + 55.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.9T + 1.21e4T^{2} \) |
| 31 | \( 1 + 281.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 390.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 29.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 9.39T + 7.95e4T^{2} \) |
| 47 | \( 1 + 469.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 607.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 523.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 854.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 149.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 462.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 844.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 91.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 671.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 211.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−8.667292030565916140117161765266, −7.69531737888007766391260531705, −6.90623280519045130582474639854, −5.97359872202995393617385114455, −5.53435663559863928497869452793, −4.83931138687408654591255118549, −3.85911383208860615814155361991, −3.13881226276043767324723843213, −2.17320708254868037060966983748, −0.51305280189833957909358177687,
0.51305280189833957909358177687, 2.17320708254868037060966983748, 3.13881226276043767324723843213, 3.85911383208860615814155361991, 4.83931138687408654591255118549, 5.53435663559863928497869452793, 5.97359872202995393617385114455, 6.90623280519045130582474639854, 7.69531737888007766391260531705, 8.667292030565916140117161765266
