| L(s) = 1 | + 2.63·2-s − 3.04·3-s + 4.91·4-s − 5-s − 8.00·6-s − 0.574·7-s + 7.67·8-s + 6.26·9-s − 2.63·10-s + 2.57·11-s − 14.9·12-s + 0.468·13-s − 1.51·14-s + 3.04·15-s + 10.3·16-s − 4.08·17-s + 16.4·18-s − 4.91·20-s + 1.74·21-s + 6.77·22-s + 1.51·23-s − 23.3·24-s + 25-s + 1.23·26-s − 9.92·27-s − 2.82·28-s + 4.08·29-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 1.75·3-s + 2.45·4-s − 0.447·5-s − 3.26·6-s − 0.217·7-s + 2.71·8-s + 2.08·9-s − 0.831·10-s + 0.776·11-s − 4.31·12-s + 0.129·13-s − 0.403·14-s + 0.785·15-s + 2.58·16-s − 0.991·17-s + 3.88·18-s − 1.09·20-s + 0.381·21-s + 1.44·22-s + 0.315·23-s − 4.76·24-s + 0.200·25-s + 0.241·26-s − 1.90·27-s − 0.534·28-s + 0.758·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.101632804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.101632804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 + 0.574T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 0.468T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 + 0.574T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 + 6.66T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−9.638649315189714397813310519935, −8.128397823031934587517722169541, −6.89589188244155168653555171024, −6.61054074024909909076916988385, −5.99335280573231450040210584931, −5.09483183582300447916141946734, −4.45874657319188368051399772900, −3.89502231656176158638969992703, −2.57734648814662431764233080693, −1.04579847525429894475470299308,
1.04579847525429894475470299308, 2.57734648814662431764233080693, 3.89502231656176158638969992703, 4.45874657319188368051399772900, 5.09483183582300447916141946734, 5.99335280573231450040210584931, 6.61054074024909909076916988385, 6.89589188244155168653555171024, 8.128397823031934587517722169541, 9.638649315189714397813310519935
