| L(s) = 1 | + 2.72·2-s − 0.598·4-s − 19.6·5-s − 3.08·7-s − 23.3·8-s − 53.4·10-s + 32.9·11-s + 16.6·13-s − 8.38·14-s − 58.8·16-s + 16.1·17-s + 67.4·19-s + 11.7·20-s + 89.6·22-s − 100.·23-s + 260.·25-s + 45.2·26-s + 1.84·28-s + 78.4·29-s + 205.·31-s + 27.0·32-s + 43.9·34-s + 60.5·35-s + 222.·37-s + 183.·38-s + 459.·40-s + 153.·41-s + ⋯ |
| L(s) = 1 | + 0.961·2-s − 0.0748·4-s − 1.75·5-s − 0.166·7-s − 1.03·8-s − 1.69·10-s + 0.903·11-s + 0.354·13-s − 0.160·14-s − 0.919·16-s + 0.230·17-s + 0.815·19-s + 0.131·20-s + 0.868·22-s − 0.906·23-s + 2.08·25-s + 0.341·26-s + 0.0124·28-s + 0.502·29-s + 1.19·31-s + 0.149·32-s + 0.221·34-s + 0.292·35-s + 0.986·37-s + 0.783·38-s + 1.81·40-s + 0.585·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 2.72T + 8T^{2} \) |
| 5 | \( 1 + 19.6T + 125T^{2} \) |
| 7 | \( 1 + 3.08T + 343T^{2} \) |
| 11 | \( 1 - 32.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 153.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 59.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 19.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 757.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 299.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 610.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 204.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 46.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 189.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 988.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 781.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 349.T + 9.12e5T^{2} \) |
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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
−8.182008389554983662294249719460, −7.64567978254809956066344344784, −6.58910778556060295839017311526, −5.97505541351326690857987240816, −4.73308904532510926471835429854, −4.30118073290717643587788165467, −3.52566823205473408938718276120, −2.94538922665443031438219768479, −1.07127770434894504720683096963, 0,
1.07127770434894504720683096963, 2.94538922665443031438219768479, 3.52566823205473408938718276120, 4.30118073290717643587788165467, 4.73308904532510926471835429854, 5.97505541351326690857987240816, 6.58910778556060295839017311526, 7.64567978254809956066344344784, 8.182008389554983662294249719460
