| L(s) = 1 | + 4.78·2-s + 9.82·3-s + 14.8·4-s − 5·5-s + 46.9·6-s − 5.33·7-s + 32.8·8-s + 69.5·9-s − 23.9·10-s + 11·11-s + 146.·12-s + 59.8·13-s − 25.5·14-s − 49.1·15-s + 38.1·16-s − 94.3·17-s + 332.·18-s + 19·19-s − 74.3·20-s − 52.3·21-s + 52.6·22-s − 32.5·23-s + 322.·24-s + 25·25-s + 286.·26-s + 417.·27-s − 79.2·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.89·3-s + 1.85·4-s − 0.447·5-s + 3.19·6-s − 0.287·7-s + 1.45·8-s + 2.57·9-s − 0.756·10-s + 0.301·11-s + 3.51·12-s + 1.27·13-s − 0.486·14-s − 0.845·15-s + 0.596·16-s − 1.34·17-s + 4.35·18-s + 0.229·19-s − 0.831·20-s − 0.544·21-s + 0.509·22-s − 0.295·23-s + 2.74·24-s + 0.200·25-s + 2.15·26-s + 2.97·27-s − 0.535·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.27149307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.27149307\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 4.78T + 8T^{2} \) |
| 3 | \( 1 - 9.82T + 27T^{2} \) |
| 7 | \( 1 + 5.33T + 343T^{2} \) |
| 13 | \( 1 - 59.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 32.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 43.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 373.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 155.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 287.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 480.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 199.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 526.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 423.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 556.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 591.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 247.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 596.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 512.T + 9.12e5T^{2} \) |
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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
−9.327045090117752966130968345482, −8.605412444236808343463750565304, −7.86524925127871234711713622338, −6.80167670608860182453731615341, −6.27866133368060821478148486638, −4.65238338293005865153902093254, −4.12487223257024634011526871059, −3.30647461351338829206090146374, −2.69882473397428481755761538444, −1.55218266725475711973112077695,
1.55218266725475711973112077695, 2.69882473397428481755761538444, 3.30647461351338829206090146374, 4.12487223257024634011526871059, 4.65238338293005865153902093254, 6.27866133368060821478148486638, 6.80167670608860182453731615341, 7.86524925127871234711713622338, 8.605412444236808343463750565304, 9.327045090117752966130968345482
