| L(s) = 1 | + 2.65·2-s − 1.26·3-s + 5.05·4-s − 1.00·5-s − 3.34·6-s + 4.25·7-s + 8.10·8-s − 1.41·9-s − 2.65·10-s + 0.380·11-s − 6.36·12-s − 13-s + 11.3·14-s + 1.26·15-s + 11.4·16-s + 1.78·17-s − 3.74·18-s + 5.94·19-s − 5.05·20-s − 5.36·21-s + 1.01·22-s − 2.85·23-s − 10.2·24-s − 3.99·25-s − 2.65·26-s + 5.55·27-s + 21.5·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 0.727·3-s + 2.52·4-s − 0.447·5-s − 1.36·6-s + 1.60·7-s + 2.86·8-s − 0.470·9-s − 0.839·10-s + 0.114·11-s − 1.83·12-s − 0.277·13-s + 3.02·14-s + 0.325·15-s + 2.85·16-s + 0.431·17-s − 0.883·18-s + 1.36·19-s − 1.12·20-s − 1.17·21-s + 0.215·22-s − 0.596·23-s − 2.08·24-s − 0.799·25-s − 0.520·26-s + 1.07·27-s + 4.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.581622544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.581622544\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 - 0.380T + 11T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 - 4.62T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 - 0.532T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 - 7.08T + 71T^{2} \) |
| 73 | \( 1 + 9.75T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 5.36T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−10.07291236636057015819888209314, −8.426724692540751563736959167755, −7.68869029767127652962330634193, −6.93275607104443478121172433692, −5.83797006620140796125488294267, −5.30291873004963004872129340056, −4.70452010482651293533908809654, −3.81803921637804729217522642903, −2.73044677156474455121237483842, −1.46717235782961165150659372722,
1.46717235782961165150659372722, 2.73044677156474455121237483842, 3.81803921637804729217522642903, 4.70452010482651293533908809654, 5.30291873004963004872129340056, 5.83797006620140796125488294267, 6.93275607104443478121172433692, 7.68869029767127652962330634193, 8.426724692540751563736959167755, 10.07291236636057015819888209314
