| L(s) = 1 | + 2-s + 0.108·3-s + 4-s − 1.47·5-s + 0.108·6-s − 7-s + 8-s − 2.98·9-s − 1.47·10-s + 3.35·11-s + 0.108·12-s + 1.08·13-s − 14-s − 0.160·15-s + 16-s + 0.00285·17-s − 2.98·18-s − 1.47·20-s − 0.108·21-s + 3.35·22-s + 1.10·23-s + 0.108·24-s − 2.80·25-s + 1.08·26-s − 0.651·27-s − 28-s − 3.77·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0628·3-s + 0.5·4-s − 0.661·5-s + 0.0444·6-s − 0.377·7-s + 0.353·8-s − 0.996·9-s − 0.467·10-s + 1.01·11-s + 0.0314·12-s + 0.300·13-s − 0.267·14-s − 0.0415·15-s + 0.250·16-s + 0.000692·17-s − 0.704·18-s − 0.330·20-s − 0.0237·21-s + 0.716·22-s + 0.230·23-s + 0.0222·24-s − 0.561·25-s + 0.212·26-s − 0.125·27-s − 0.188·28-s − 0.701·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.108T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 - 0.00285T + 17T^{2} \) |
| 23 | \( 1 - 1.10T + 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 + 0.689T + 31T^{2} \) |
| 37 | \( 1 + 7.01T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 9.33T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 1.98T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 + 3.29T + 61T^{2} \) |
| 67 | \( 1 - 1.13T + 67T^{2} \) |
| 71 | \( 1 + 9.20T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 - 8.24T + 83T^{2} \) |
| 89 | \( 1 - 0.669T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−7.79636962353007655309628294495, −7.05036838716069801806392212742, −6.34162213979973847032387811150, −5.72608817628507192182744009847, −4.91821177451930486693282489983, −3.90390710291417826457541748366, −3.54379697724643340091556050364, −2.64175511537901909327376510859, −1.49271060918704932929773731450, 0,
1.49271060918704932929773731450, 2.64175511537901909327376510859, 3.54379697724643340091556050364, 3.90390710291417826457541748366, 4.91821177451930486693282489983, 5.72608817628507192182744009847, 6.34162213979973847032387811150, 7.05036838716069801806392212742, 7.79636962353007655309628294495
