| L(s) = 1 | − 2.82i·2-s − 5.19·3-s − 8.00·4-s − 19.4·5-s + 14.6i·6-s + 2.40·7-s + 22.6i·8-s + 27·9-s + 55.0i·10-s − 210. i·11-s + 41.5·12-s − 188. i·13-s − 6.79i·14-s + 101.·15-s + 64.0·16-s + 466.·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.500·4-s − 0.777·5-s + 0.408i·6-s + 0.0490·7-s + 0.353i·8-s + 0.333·9-s + 0.550i·10-s − 1.73i·11-s + 0.288·12-s − 1.11i·13-s − 0.0346i·14-s + 0.449·15-s + 0.250·16-s + 1.61·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3631438723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3631438723\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (-3.03e3 + 1.70e3i)T \) |
| good | 5 | \( 1 + 19.4T + 625T^{2} \) |
| 7 | \( 1 - 2.40T + 2.40e3T^{2} \) |
| 11 | \( 1 + 210. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 188. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 466.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 69.9T + 1.30e5T^{2} \) |
| 23 | \( 1 - 163. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 850.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.50e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 440. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.80e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.85e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.06e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.32e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 53.5iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.43e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.67e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 1.13e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 7.66e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.04e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 657. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 126. iT - 8.85e7T^{2} \) |
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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
−10.42608413716827549809425031427, −9.543858928441890259417834282922, −8.179797680310181833704540645484, −7.72176811682349444074742437876, −5.98565276274510604360399259635, −5.32460983911299535776204440247, −3.81322548608205142236852999952, −3.09349109042728867826029068617, −1.08952674825127697595839180070, −0.13912281654903514774511781285,
1.62707065616197271948784460833, 3.72179821929718721957913903191, 4.65951933999664857786065065470, 5.58579441433687429733936815331, 7.03974567348685096167645989780, 7.28695730263779788140334676469, 8.508561068131895983985250533291, 9.663056443186841257804397179635, 10.35382607184093103758093129933, 11.83145119832531961891401748470
