| L(s) = 1 | − 2.00·2-s − 1.75·3-s + 2.03·4-s − 0.905·5-s + 3.53·6-s − 0.0686·8-s + 0.0942·9-s + 1.81·10-s + 0.716·11-s − 3.57·12-s − 13-s + 1.59·15-s − 3.93·16-s − 2.35·17-s − 0.189·18-s − 6.63·19-s − 1.84·20-s − 1.43·22-s + 3.75·23-s + 0.120·24-s − 4.17·25-s + 2.00·26-s + 5.11·27-s + 3.25·29-s − 3.20·30-s − 1.57·31-s + 8.03·32-s + ⋯ |
| L(s) = 1 | − 1.42·2-s − 1.01·3-s + 1.01·4-s − 0.405·5-s + 1.44·6-s − 0.0242·8-s + 0.0314·9-s + 0.575·10-s + 0.215·11-s − 1.03·12-s − 0.277·13-s + 0.411·15-s − 0.982·16-s − 0.570·17-s − 0.0446·18-s − 1.52·19-s − 0.411·20-s − 0.306·22-s + 0.783·23-s + 0.0246·24-s − 0.835·25-s + 0.393·26-s + 0.983·27-s + 0.604·29-s − 0.584·30-s − 0.282·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3060607450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3060607450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.00T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 0.905T + 5T^{2} \) |
| 11 | \( 1 - 0.716T + 11T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 - 5.20T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 - 8.31T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 0.716T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.39T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 3.47T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 7.43T + 97T^{2} \) |
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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
−10.57323558578137676195984937419, −9.793600687529902077373433257744, −8.738217503189565302578205662201, −8.227241499656225880955860428158, −7.03542766398972229338840233414, −6.45151260537239723045933174400, −5.19146952663274279601492832562, −4.11950990912236128083857857925, −2.25130907375641470428688420174, −0.59997345471379059232210335297,
0.59997345471379059232210335297, 2.25130907375641470428688420174, 4.11950990912236128083857857925, 5.19146952663274279601492832562, 6.45151260537239723045933174400, 7.03542766398972229338840233414, 8.227241499656225880955860428158, 8.738217503189565302578205662201, 9.793600687529902077373433257744, 10.57323558578137676195984937419
