| L(s) = 1 | + 0.554·2-s + 0.198·3-s − 1.69·4-s + 2·5-s + 0.109·6-s + 2.55·7-s − 2.04·8-s − 2.96·9-s + 1.10·10-s + 4.74·11-s − 0.335·12-s + 0.643·13-s + 1.41·14-s + 0.396·15-s + 2.24·16-s + 1.10·17-s − 1.64·18-s − 5.35·19-s − 3.38·20-s + 0.506·21-s + 2.63·22-s + 5.24·23-s − 0.405·24-s − 25-s + 0.356·26-s − 1.18·27-s − 4.32·28-s + ⋯ |
| L(s) = 1 | + 0.392·2-s + 0.114·3-s − 0.846·4-s + 0.894·5-s + 0.0448·6-s + 0.965·7-s − 0.724·8-s − 0.986·9-s + 0.350·10-s + 1.42·11-s − 0.0967·12-s + 0.178·13-s + 0.378·14-s + 0.102·15-s + 0.561·16-s + 0.269·17-s − 0.387·18-s − 1.22·19-s − 0.756·20-s + 0.110·21-s + 0.560·22-s + 1.09·23-s − 0.0828·24-s − 0.200·25-s + 0.0699·26-s − 0.227·27-s − 0.816·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346873420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346873420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 0.554T + 2T^{2} \) |
| 3 | \( 1 - 0.198T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 0.643T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 - 0.911T + 29T^{2} \) |
| 31 | \( 1 - 5.96T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 7.76T + 41T^{2} \) |
| 47 | \( 1 + T + 47T^{2} \) |
| 53 | \( 1 - 7.74T + 53T^{2} \) |
| 59 | \( 1 + 8.33T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 - 0.466T + 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−9.123447871982721538867726660235, −8.611379768291748933683547851190, −7.944646376909752397980541896852, −6.49880214036193926173458162137, −6.02691715588573002255056218078, −5.10175393217549823851051083036, −4.42234383831074363223197321373, −3.45544998258703965573948008112, −2.28472745983513934267431981290, −1.05083202086435359206324429996,
1.05083202086435359206324429996, 2.28472745983513934267431981290, 3.45544998258703965573948008112, 4.42234383831074363223197321373, 5.10175393217549823851051083036, 6.02691715588573002255056218078, 6.49880214036193926173458162137, 7.944646376909752397980541896852, 8.611379768291748933683547851190, 9.123447871982721538867726660235
