| L(s) = 1 | − 2.07·2-s − 3.01·3-s + 2.29·4-s − 2.69·5-s + 6.24·6-s + 7-s − 0.602·8-s + 6.09·9-s + 5.57·10-s + 1.66·11-s − 6.90·12-s − 2.07·14-s + 8.12·15-s − 3.33·16-s + 6.90·17-s − 12.6·18-s − 7.92·19-s − 6.16·20-s − 3.01·21-s − 3.43·22-s + 1.95·23-s + 1.81·24-s + 2.24·25-s − 9.34·27-s + 2.29·28-s − 2.71·29-s − 16.8·30-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 1.74·3-s + 1.14·4-s − 1.20·5-s + 2.55·6-s + 0.377·7-s − 0.212·8-s + 2.03·9-s + 1.76·10-s + 0.500·11-s − 1.99·12-s − 0.553·14-s + 2.09·15-s − 0.833·16-s + 1.67·17-s − 2.97·18-s − 1.81·19-s − 1.37·20-s − 0.658·21-s − 0.733·22-s + 0.406·23-s + 0.370·24-s + 0.449·25-s − 1.79·27-s + 0.432·28-s − 0.504·29-s − 3.07·30-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
| \(L(1)\) |
\(\approx\) |
\(0.2308920690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2308920690\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.07T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 + 7.08T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 4.52T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 + 1.27T + 67T^{2} \) |
| 71 | \( 1 + 2.93T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 2.91T + 79T^{2} \) |
| 83 | \( 1 + 2.33T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−10.09704367428878304838547469245, −8.864725194302782535326011701280, −8.142468839393243104938156385460, −7.30878770889286936142506588080, −6.75835675766370099580488191172, −5.69205663366576325980714280772, −4.68927275158315843895273156828, −3.80027808470737391379587990523, −1.62725501183240577449120536261, −0.50967539633893059922319959070,
0.50967539633893059922319959070, 1.62725501183240577449120536261, 3.80027808470737391379587990523, 4.68927275158315843895273156828, 5.69205663366576325980714280772, 6.75835675766370099580488191172, 7.30878770889286936142506588080, 8.142468839393243104938156385460, 8.864725194302782535326011701280, 10.09704367428878304838547469245
