| L(s) = 1 | + 2-s + 3-s + 4-s + 3.23·5-s + 6-s − 4.47·7-s + 8-s + 9-s + 3.23·10-s + 0.763·11-s + 12-s − 4.47·13-s − 4.47·14-s + 3.23·15-s + 16-s + 4·17-s + 18-s + 7.70·19-s + 3.23·20-s − 4.47·21-s + 0.763·22-s + 24-s + 5.47·25-s − 4.47·26-s + 27-s − 4.47·28-s + 4.47·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.44·5-s + 0.408·6-s − 1.69·7-s + 0.353·8-s + 0.333·9-s + 1.02·10-s + 0.230·11-s + 0.288·12-s − 1.24·13-s − 1.19·14-s + 0.835·15-s + 0.250·16-s + 0.970·17-s + 0.235·18-s + 1.76·19-s + 0.723·20-s − 0.975·21-s + 0.162·22-s + 0.204·24-s + 1.09·25-s − 0.877·26-s + 0.192·27-s − 0.845·28-s + 0.830·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.216949575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.216949575\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9.23T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 0.763T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.961962908553550004489310627762, −7.68009652823819855391609531230, −7.00740905512036870265019794104, −6.35508843535325978501003240906, −5.62210841135063879684587788492, −5.00481801069111586271087232843, −3.75483423974099476760505675854, −2.91139738691773074664742373955, −2.51716272798110563496820644548, −1.15059298584104255275817229374,
1.15059298584104255275817229374, 2.51716272798110563496820644548, 2.91139738691773074664742373955, 3.75483423974099476760505675854, 5.00481801069111586271087232843, 5.62210841135063879684587788492, 6.35508843535325978501003240906, 7.00740905512036870265019794104, 7.68009652823819855391609531230, 8.961962908553550004489310627762
