| L(s) = 1 | + 5.18·2-s + 18.8·4-s − 5.12·5-s + 5.19·7-s + 56.2·8-s − 26.5·10-s + 55.2·11-s + 76.1·13-s + 26.9·14-s + 140.·16-s + 20.2·17-s − 39.8·19-s − 96.5·20-s + 286.·22-s − 73.2·23-s − 98.7·25-s + 394.·26-s + 97.9·28-s + 183.·29-s − 177.·31-s + 278.·32-s + 104.·34-s − 26.6·35-s + 199.·37-s − 206.·38-s − 288.·40-s + 359.·41-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 2.35·4-s − 0.458·5-s + 0.280·7-s + 2.48·8-s − 0.839·10-s + 1.51·11-s + 1.62·13-s + 0.514·14-s + 2.19·16-s + 0.288·17-s − 0.481·19-s − 1.07·20-s + 2.77·22-s − 0.663·23-s − 0.790·25-s + 2.97·26-s + 0.661·28-s + 1.17·29-s − 1.03·31-s + 1.54·32-s + 0.528·34-s − 0.128·35-s + 0.888·37-s − 0.881·38-s − 1.13·40-s + 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.057736051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.057736051\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 5.18T + 8T^{2} \) |
| 5 | \( 1 + 5.12T + 125T^{2} \) |
| 7 | \( 1 - 5.19T + 343T^{2} \) |
| 11 | \( 1 - 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 359.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 288.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 42.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 623.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 943.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 162.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 524.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 937.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 801.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 133.T + 9.12e5T^{2} \) |
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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
−8.501342192404878137339408110664, −7.76732099669563480449495095063, −6.68711937876496815939526689403, −6.26927107148577007582771883951, −5.55052387575413445335638911560, −4.42563027369943145813381268021, −3.91845939128271239179560239028, −3.37335366069317403441920853593, −2.07231209120825873706252347204, −1.13333970288119656882268983367,
1.13333970288119656882268983367, 2.07231209120825873706252347204, 3.37335366069317403441920853593, 3.91845939128271239179560239028, 4.42563027369943145813381268021, 5.55052387575413445335638911560, 6.26927107148577007582771883951, 6.68711937876496815939526689403, 7.76732099669563480449495095063, 8.501342192404878137339408110664
