| L(s) = 1 | − 2.34·2-s + 2.86·3-s + 3.51·4-s − 5-s − 6.73·6-s + 3.38·7-s − 3.56·8-s + 5.21·9-s + 2.34·10-s + 10.0·12-s − 1.48·13-s − 7.94·14-s − 2.86·15-s + 1.33·16-s + 3.73·17-s − 12.2·18-s + 3.21·19-s − 3.51·20-s + 9.69·21-s − 2.51·23-s − 10.2·24-s + 25-s + 3.48·26-s + 6.34·27-s + 11.9·28-s − 4.69·29-s + 6.73·30-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.65·3-s + 1.75·4-s − 0.447·5-s − 2.74·6-s + 1.27·7-s − 1.26·8-s + 1.73·9-s + 0.742·10-s + 2.91·12-s − 0.411·13-s − 2.12·14-s − 0.740·15-s + 0.334·16-s + 0.905·17-s − 2.88·18-s + 0.737·19-s − 0.786·20-s + 2.11·21-s − 0.524·23-s − 2.08·24-s + 0.200·25-s + 0.683·26-s + 1.22·27-s + 2.24·28-s − 0.872·29-s + 1.22·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292913420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292913420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 2.86T + 3T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 + 2.69T + 37T^{2} \) |
| 41 | \( 1 + 4.55T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 7.83T + 67T^{2} \) |
| 71 | \( 1 - 2.51T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 + 7.55T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.07247177475838222910559155654, −9.750440621980520096414692078670, −8.630777768741642911386832911604, −8.162802840504624379963994081722, −7.73487137296792916089105770616, −6.90075166667405528443125815046, −4.97740484161791914639572963685, −3.59261515728408605424602592034, −2.36553984896459084860831827564, −1.35847909486953709471518764152,
1.35847909486953709471518764152, 2.36553984896459084860831827564, 3.59261515728408605424602592034, 4.97740484161791914639572963685, 6.90075166667405528443125815046, 7.73487137296792916089105770616, 8.162802840504624379963994081722, 8.630777768741642911386832911604, 9.750440621980520096414692078670, 10.07247177475838222910559155654
