| L(s) = 1 | + 5i·3-s − 26i·7-s + 2·9-s − 11·11-s − 52i·13-s + 46i·17-s + 96·19-s + 130·21-s − 27i·23-s + 145i·27-s − 16·29-s − 293·31-s − 55i·33-s − 29i·37-s + 260·39-s + ⋯ |
| L(s) = 1 | + 0.962i·3-s − 1.40i·7-s + 0.0740·9-s − 0.301·11-s − 1.10i·13-s + 0.656i·17-s + 1.15·19-s + 1.35·21-s − 0.244i·23-s + 1.03i·27-s − 0.102·29-s − 1.69·31-s − 0.290i·33-s − 0.128i·37-s + 1.06·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8895199104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8895199104\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 5iT - 27T^{2} \) |
| 7 | \( 1 + 26iT - 343T^{2} \) |
| 13 | \( 1 + 52iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 46iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 96T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 16T + 2.43e4T^{2} \) |
| 31 | \( 1 + 293T + 2.97e4T^{2} \) |
| 37 | \( 1 + 29iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 472T + 6.89e4T^{2} \) |
| 43 | \( 1 - 110iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 224iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 754iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 825T + 2.05e5T^{2} \) |
| 61 | \( 1 + 548T + 2.26e5T^{2} \) |
| 67 | \( 1 + 123iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 526T + 4.93e5T^{2} \) |
| 83 | \( 1 - 158iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 263iT - 9.12e5T^{2} \) |
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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
−9.518185590058105070734044733722, −8.353165908936486322121706033073, −7.55481787992522896901098235085, −6.84943655516385004742150709241, −5.52203741109500171206823257816, −4.84174649087024287585206980256, −3.78378945416470657303080105886, −3.30123817992819492569185438372, −1.50421869892829883906914081587, −0.21201197387554962831622601733,
1.42704223631353321758901095522, 2.21983372178225673724001671261, 3.28262386514704381172763546234, 4.73768679132738349005874899839, 5.59803717537564205586337703883, 6.44213279972702713530604033628, 7.29987845767317130264233319873, 7.912737186897536774101035289060, 9.192621601554903161167723725784, 9.269672765884826741450980508113
