L(s) = 1 | − 2.02e16·2-s − 7.88e25·3-s − 2.40e32·4-s + 1.60e38·5-s + 1.59e42·6-s + 1.73e45·7-s + 1.79e49·8-s − 3.92e51·9-s − 3.24e54·10-s − 1.26e56·11-s + 1.89e58·12-s − 6.03e60·13-s − 3.50e61·14-s − 1.26e64·15-s − 2.07e65·16-s − 1.23e67·17-s + 7.93e67·18-s + 8.35e69·19-s − 3.86e70·20-s − 1.36e71·21-s + 2.55e72·22-s + 1.30e74·23-s − 1.41e75·24-s + 1.04e76·25-s + 1.21e77·26-s + 1.10e78·27-s − 4.17e77·28-s + ⋯ |
L(s) = 1 | − 0.793·2-s − 0.782·3-s − 0.370·4-s + 1.29·5-s + 0.620·6-s + 0.151·7-s + 1.08·8-s − 0.387·9-s − 1.02·10-s − 0.221·11-s + 0.290·12-s − 1.17·13-s − 0.120·14-s − 1.01·15-s − 0.491·16-s − 1.07·17-s + 0.307·18-s + 1.69·19-s − 0.480·20-s − 0.118·21-s + 0.175·22-s + 0.797·23-s − 0.851·24-s + 0.676·25-s + 0.933·26-s + 1.08·27-s − 0.0563·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(110-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+54.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(55)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{111}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 + 2.02e16T + 6.49e32T^{2} \) |
| 3 | \( 1 + 7.88e25T + 1.01e52T^{2} \) |
| 5 | \( 1 - 1.60e38T + 1.54e76T^{2} \) |
| 7 | \( 1 - 1.73e45T + 1.30e92T^{2} \) |
| 11 | \( 1 + 1.26e56T + 3.24e113T^{2} \) |
| 13 | \( 1 + 6.03e60T + 2.62e121T^{2} \) |
| 17 | \( 1 + 1.23e67T + 1.31e134T^{2} \) |
| 19 | \( 1 - 8.35e69T + 2.42e139T^{2} \) |
| 23 | \( 1 - 1.30e74T + 2.68e148T^{2} \) |
| 29 | \( 1 + 1.12e79T + 2.51e159T^{2} \) |
| 31 | \( 1 + 1.73e81T + 3.61e162T^{2} \) |
| 37 | \( 1 - 3.88e85T + 8.58e170T^{2} \) |
| 41 | \( 1 - 7.86e87T + 6.21e175T^{2} \) |
| 43 | \( 1 - 3.55e88T + 1.11e178T^{2} \) |
| 47 | \( 1 + 2.02e90T + 1.81e182T^{2} \) |
| 53 | \( 1 - 1.36e94T + 8.83e187T^{2} \) |
| 59 | \( 1 - 1.85e96T + 1.05e193T^{2} \) |
| 61 | \( 1 + 1.53e97T + 3.98e194T^{2} \) |
| 67 | \( 1 + 1.76e99T + 1.10e199T^{2} \) |
| 71 | \( 1 - 1.19e101T + 6.12e201T^{2} \) |
| 73 | \( 1 + 5.70e101T + 1.26e203T^{2} \) |
| 79 | \( 1 + 3.23e103T + 6.93e206T^{2} \) |
| 83 | \( 1 + 7.45e104T + 1.51e209T^{2} \) |
| 89 | \( 1 - 2.08e106T + 3.04e212T^{2} \) |
| 97 | \( 1 + 1.80e107T + 3.61e216T^{2} \) |
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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
−13.06752390717417469834278451782, −11.24359784541137692329556531130, −9.943655614927464452992610416443, −9.058679387804406145567211830831, −7.32208622996020121582320349393, −5.70089867431840766010910606129, −4.81944974173765843683828084456, −2.49755491621002396002902594511, −1.13876916935493967162061779737, 0,
1.13876916935493967162061779737, 2.49755491621002396002902594511, 4.81944974173765843683828084456, 5.70089867431840766010910606129, 7.32208622996020121582320349393, 9.058679387804406145567211830831, 9.943655614927464452992610416443, 11.24359784541137692329556531130, 13.06752390717417469834278451782