| L(s) = 1 | − 2.22·2-s + 0.549·3-s + 2.92·4-s + 4.22·5-s − 1.22·6-s + 7-s − 2.06·8-s − 2.69·9-s − 9.36·10-s + 0.549·11-s + 1.60·12-s − 2.22·14-s + 2.31·15-s − 1.28·16-s − 2.37·17-s + 5.98·18-s + 3.61·19-s + 12.3·20-s + 0.549·21-s − 1.22·22-s + 5.81·23-s − 1.13·24-s + 12.8·25-s − 3.13·27-s + 2.92·28-s − 3.59·29-s − 5.14·30-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.317·3-s + 1.46·4-s + 1.88·5-s − 0.498·6-s + 0.377·7-s − 0.728·8-s − 0.899·9-s − 2.96·10-s + 0.165·11-s + 0.464·12-s − 0.593·14-s + 0.598·15-s − 0.320·16-s − 0.576·17-s + 1.41·18-s + 0.828·19-s + 2.76·20-s + 0.119·21-s − 0.260·22-s + 1.21·23-s − 0.231·24-s + 2.56·25-s − 0.602·27-s + 0.553·28-s − 0.668·29-s − 0.939·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227673813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227673813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 - 0.549T + 3T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 11 | \( 1 - 0.549T + 11T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 + 3.59T + 29T^{2} \) |
| 31 | \( 1 - 5.14T + 31T^{2} \) |
| 37 | \( 1 - 0.329T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 - 0.609T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 4.90T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 5.73T + 83T^{2} \) |
| 89 | \( 1 - 7.46T + 89T^{2} \) |
| 97 | \( 1 - 6.84T + 97T^{2} \) |
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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
−9.491115056715396856396705337558, −9.108767814570953696621446192785, −8.491192992397276269358661579108, −7.49743662118400457047815610451, −6.59287318291302359132335014713, −5.80179938392955000251495449137, −4.87399806217164309119971140695, −2.90230179977943845554811501593, −2.12403285274104383646148875801, −1.10608200224633273327717710853,
1.10608200224633273327717710853, 2.12403285274104383646148875801, 2.90230179977943845554811501593, 4.87399806217164309119971140695, 5.80179938392955000251495449137, 6.59287318291302359132335014713, 7.49743662118400457047815610451, 8.491192992397276269358661579108, 9.108767814570953696621446192785, 9.491115056715396856396705337558
