| L(s) = 1 | + (0.540 + 0.841i)2-s + (−0.549 − 1.64i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (1.08 − 1.35i)6-s + (−0.315 + 0.273i)7-s + (−0.989 + 0.142i)8-s + (−2.39 + 1.80i)9-s + (−0.755 − 0.654i)10-s + (1.39 + 0.897i)11-s + (1.72 + 0.182i)12-s + (−0.500 + 0.577i)13-s + (−0.400 − 0.117i)14-s + (0.989 + 1.42i)15-s + (−0.654 − 0.755i)16-s + (3.26 + 7.15i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (−0.317 − 0.948i)3-s + (−0.207 + 0.454i)4-s + (−0.429 + 0.125i)5-s + (0.442 − 0.551i)6-s + (−0.119 + 0.103i)7-s + (−0.349 + 0.0503i)8-s + (−0.798 + 0.601i)9-s + (−0.238 − 0.207i)10-s + (0.421 + 0.270i)11-s + (0.497 + 0.0527i)12-s + (−0.138 + 0.160i)13-s + (−0.107 − 0.0314i)14-s + (0.255 + 0.367i)15-s + (−0.163 − 0.188i)16-s + (0.792 + 1.73i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0489 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0489 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.891897 + 0.849261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.891897 + 0.849261i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 3 | \( 1 + (0.549 + 1.64i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-0.640 - 4.75i)T \) |
| good | 7 | \( 1 + (0.315 - 0.273i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.39 - 0.897i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.500 - 0.577i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.26 - 7.15i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.37 - 1.08i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.07 - 1.40i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.312 + 2.17i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.803 + 2.73i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.210 - 0.717i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-6.44 - 0.926i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 5.45iT - 47T^{2} \) |
| 53 | \( 1 + (0.203 + 0.234i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (8.11 + 7.03i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (10.9 - 1.56i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-7.75 - 12.0i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.35 - 5.21i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.36 - 5.17i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.95 + 3.42i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (7.61 + 2.23i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.35 + 9.41i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.70 + 5.80i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−10.95346130671142379757051803226, −9.714489450224069200010834327316, −8.649186615819116527542145594389, −7.72429609306925593976651716337, −7.28075334134603690822472732540, −6.13805577328975185735478493651, −5.63760556080154505132418771926, −4.27100050411942623760972120478, −3.17226224514850003531973664701, −1.54623183773973763049277351443,
0.63496414770437872389482672318, 2.82400074015172329742176673653, 3.65321839731700887852129706083, 4.71231448011471129133519719393, 5.36310087866067625883647720988, 6.51062567239236481235109152342, 7.67977175745969061465514193150, 8.948291093029194927600222540345, 9.501740596551772312120353545652, 10.37238361708048815614263387252
