| L(s) = 1 | − 1.64·3-s − 3.30·5-s − 3.78·7-s − 0.302·9-s − 5.42·11-s − 4.90·13-s + 5.42·15-s + 1.64·19-s + 6.21·21-s − 3.78·23-s + 5.90·25-s + 5.42·27-s − 3.69·29-s + 5.42·31-s + 8.90·33-s + 12.4·35-s − 4·37-s + 8.06·39-s + 9.21·41-s − 1.14·43-s + 1.00·45-s + 11.9·47-s + 7.30·49-s − 12.9·53-s + 17.9·55-s − 2.69·57-s + 15.1·59-s + ⋯ |
| L(s) = 1 | − 0.948·3-s − 1.47·5-s − 1.42·7-s − 0.100·9-s − 1.63·11-s − 1.36·13-s + 1.40·15-s + 0.376·19-s + 1.35·21-s − 0.788·23-s + 1.18·25-s + 1.04·27-s − 0.686·29-s + 0.974·31-s + 1.55·33-s + 2.11·35-s − 0.657·37-s + 1.29·39-s + 1.43·41-s − 0.174·43-s + 0.149·45-s + 1.74·47-s + 1.04·49-s − 1.77·53-s + 2.41·55-s − 0.357·57-s + 1.96·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 - 5.42T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 9.21T + 41T^{2} \) |
| 43 | \( 1 + 1.14T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 - 0.211T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 - 2.21T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.21T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 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
−7.39580505436222981183995667279, −6.77309591712058684280107387048, −5.91598812472226407324568377014, −5.36673037541556306932745547091, −4.64503586958196760077577392629, −3.89651128502549723720851248785, −2.98059807789981715139826328789, −2.53140187540203905132814020557, −0.53896359548457865668962660861, 0,
0.53896359548457865668962660861, 2.53140187540203905132814020557, 2.98059807789981715139826328789, 3.89651128502549723720851248785, 4.64503586958196760077577392629, 5.36673037541556306932745547091, 5.91598812472226407324568377014, 6.77309591712058684280107387048, 7.39580505436222981183995667279
