| L(s) = 1 | + 2-s + 2.12·3-s + 4-s + 3.84·5-s + 2.12·6-s − 4.95·7-s + 8-s + 1.52·9-s + 3.84·10-s − 0.453·11-s + 2.12·12-s − 0.0945·13-s − 4.95·14-s + 8.18·15-s + 16-s + 1.41·17-s + 1.52·18-s − 2.49·19-s + 3.84·20-s − 10.5·21-s − 0.453·22-s − 2.25·23-s + 2.12·24-s + 9.77·25-s − 0.0945·26-s − 3.12·27-s − 4.95·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.22·3-s + 0.5·4-s + 1.71·5-s + 0.868·6-s − 1.87·7-s + 0.353·8-s + 0.509·9-s + 1.21·10-s − 0.136·11-s + 0.614·12-s − 0.0262·13-s − 1.32·14-s + 2.11·15-s + 0.250·16-s + 0.342·17-s + 0.360·18-s − 0.571·19-s + 0.859·20-s − 2.30·21-s − 0.0966·22-s − 0.470·23-s + 0.434·24-s + 1.95·25-s − 0.0185·26-s − 0.602·27-s − 0.936·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.353001232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.353001232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 + 4.95T + 7T^{2} \) |
| 11 | \( 1 + 0.453T + 11T^{2} \) |
| 13 | \( 1 + 0.0945T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.49T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 - 9.96T + 53T^{2} \) |
| 59 | \( 1 - 7.89T + 59T^{2} \) |
| 61 | \( 1 - 0.950T + 61T^{2} \) |
| 67 | \( 1 + 8.06T + 67T^{2} \) |
| 71 | \( 1 + 3.06T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 1.77T + 89T^{2} \) |
| 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
−10.42646103165960243971361188786, −9.805120184879420734040051587392, −9.324269768843518886320779473703, −8.283798994452844320220048175872, −6.87236943525028062951665485539, −6.23831475424351470791120212839, −5.38969618328243421759156854697, −3.73127925729410996263846611519, −2.87917663665581735234830298866, −2.07082619117076174479885311018,
2.07082619117076174479885311018, 2.87917663665581735234830298866, 3.73127925729410996263846611519, 5.38969618328243421759156854697, 6.23831475424351470791120212839, 6.87236943525028062951665485539, 8.283798994452844320220048175872, 9.324269768843518886320779473703, 9.805120184879420734040051587392, 10.42646103165960243971361188786
