| L(s) = 1 | − 2.32·2-s − 0.113·3-s + 3.41·4-s + 1.83·5-s + 0.264·6-s − 3.17·7-s − 3.30·8-s − 2.98·9-s − 4.26·10-s + 11-s − 0.389·12-s − 0.0911·13-s + 7.39·14-s − 0.208·15-s + 0.855·16-s − 3.04·17-s + 6.95·18-s − 0.389·19-s + 6.27·20-s + 0.361·21-s − 2.32·22-s + 1.77·23-s + 0.376·24-s − 1.63·25-s + 0.212·26-s + 0.681·27-s − 10.8·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.0657·3-s + 1.70·4-s + 0.819·5-s + 0.108·6-s − 1.20·7-s − 1.16·8-s − 0.995·9-s − 1.34·10-s + 0.301·11-s − 0.112·12-s − 0.0252·13-s + 1.97·14-s − 0.0538·15-s + 0.213·16-s − 0.739·17-s + 1.63·18-s − 0.0893·19-s + 1.40·20-s + 0.0788·21-s − 0.496·22-s + 0.369·23-s + 0.0767·24-s − 0.327·25-s + 0.0415·26-s + 0.131·27-s − 2.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 + 0.113T + 3T^{2} \) |
| 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 13 | \( 1 + 0.0911T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 0.389T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 31 | \( 1 - 4.88T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 - 5.27T + 41T^{2} \) |
| 43 | \( 1 - 6.99T + 43T^{2} \) |
| 47 | \( 1 + 3.78T + 47T^{2} \) |
| 53 | \( 1 - 5.48T + 53T^{2} \) |
| 59 | \( 1 + 4.22T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 + 0.298T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 8.91T + 97T^{2} \) |
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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
−7.39186434464748997268794843926, −6.80675577631515017163117434649, −6.17828487056950250565145244087, −5.78225981494583567427964830493, −4.65717890716718127319642899467, −3.50971505766680434056122421460, −2.63431627473701723243690629758, −2.10760716536651364064622141193, −0.935124673916360553437594558447, 0,
0.935124673916360553437594558447, 2.10760716536651364064622141193, 2.63431627473701723243690629758, 3.50971505766680434056122421460, 4.65717890716718127319642899467, 5.78225981494583567427964830493, 6.17828487056950250565145244087, 6.80675577631515017163117434649, 7.39186434464748997268794843926
