| L(s) = 1 | + 1.17·2-s − 1.53·3-s − 0.630·4-s − 4.17·5-s − 1.80·6-s + 1.07·7-s − 3.07·8-s − 0.630·9-s − 4.87·10-s − 11-s + 0.971·12-s + 5.51·13-s + 1.26·14-s + 6.41·15-s − 2.34·16-s − 17-s − 0.738·18-s − 4.58·19-s + 2.63·20-s − 1.65·21-s − 1.17·22-s − 5.53·23-s + 4.73·24-s + 12.3·25-s + 6.44·26-s + 5.58·27-s − 0.680·28-s + ⋯ |
| L(s) = 1 | + 0.827·2-s − 0.888·3-s − 0.315·4-s − 1.86·5-s − 0.735·6-s + 0.407·7-s − 1.08·8-s − 0.210·9-s − 1.54·10-s − 0.301·11-s + 0.280·12-s + 1.52·13-s + 0.337·14-s + 1.65·15-s − 0.585·16-s − 0.242·17-s − 0.173·18-s − 1.05·19-s + 0.588·20-s − 0.362·21-s − 0.249·22-s − 1.15·23-s + 0.967·24-s + 2.47·25-s + 1.26·26-s + 1.07·27-s − 0.128·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 + 5.53T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 + 6.83T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 3.95T + 53T^{2} \) |
| 59 | \( 1 - 8.31T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 + 4.53T + 73T^{2} \) |
| 79 | \( 1 - 7.80T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 + 1.18T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−12.04043380415450955393945525255, −11.38369395102381117736194412988, −10.70924119838047052921952797063, −8.705150012486581926468625520650, −8.123131049230366563520930478570, −6.58669561199221272604790698780, −5.46641883005619473066294696758, −4.32959758370978532751176829758, −3.55593028620082153062431215700, 0,
3.55593028620082153062431215700, 4.32959758370978532751176829758, 5.46641883005619473066294696758, 6.58669561199221272604790698780, 8.123131049230366563520930478570, 8.705150012486581926468625520650, 10.70924119838047052921952797063, 11.38369395102381117736194412988, 12.04043380415450955393945525255
