| L(s) = 1 | − 4.92·2-s − 4.29·3-s + 16.2·4-s + 2.55·5-s + 21.1·6-s − 40.8·8-s − 8.58·9-s − 12.5·10-s + 30.6·11-s − 69.8·12-s − 6.03·13-s − 10.9·15-s + 70.9·16-s − 70.2·17-s + 42.3·18-s − 127.·19-s + 41.5·20-s − 151.·22-s − 58.0·23-s + 175.·24-s − 118.·25-s + 29.7·26-s + 152.·27-s − 127.·29-s + 53.9·30-s − 150.·31-s − 23.0·32-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.825·3-s + 2.03·4-s + 0.228·5-s + 1.43·6-s − 1.80·8-s − 0.318·9-s − 0.397·10-s + 0.841·11-s − 1.68·12-s − 0.128·13-s − 0.188·15-s + 1.10·16-s − 1.00·17-s + 0.554·18-s − 1.53·19-s + 0.464·20-s − 1.46·22-s − 0.525·23-s + 1.49·24-s − 0.947·25-s + 0.224·26-s + 1.08·27-s − 0.815·29-s + 0.328·30-s − 0.870·31-s − 0.127·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02638363380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02638363380\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 4.92T + 8T^{2} \) |
| 3 | \( 1 + 4.29T + 27T^{2} \) |
| 5 | \( 1 - 2.55T + 125T^{2} \) |
| 11 | \( 1 - 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.03T + 2.19e3T^{2} \) |
| 17 | \( 1 + 70.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 127.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 418.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 561.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 227.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 760.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 486.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 111.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 711.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 193.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 960.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 344.T + 9.12e5T^{2} \) |
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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
−8.704393107837895874187042861452, −8.169227689370711794420798726012, −7.09031256690665859295227051951, −6.50818225344737205334559892847, −5.96371388495318910911940970474, −4.83097377216979553587800993390, −3.63334171798479055216742647556, −2.17672326720838456262366712736, −1.60113255094757585226609446210, −0.097074115408620729157808840582,
0.097074115408620729157808840582, 1.60113255094757585226609446210, 2.17672326720838456262366712736, 3.63334171798479055216742647556, 4.83097377216979553587800993390, 5.96371388495318910911940970474, 6.50818225344737205334559892847, 7.09031256690665859295227051951, 8.169227689370711794420798726012, 8.704393107837895874187042861452
