L(s) = 1 | + 3-s − 1.87·5-s + 3.28·7-s + 9-s + 4.43·13-s − 1.87·15-s + 6.72·17-s − 1.69·19-s + 3.28·21-s + 1.69·23-s − 1.47·25-s + 27-s + 9.75·29-s − 1.59·31-s − 6.16·35-s + 4.53·37-s + 4.43·39-s + 3.33·41-s − 7.16·43-s − 1.87·45-s − 4.47·47-s + 3.77·49-s + 6.72·51-s − 0.184·53-s − 1.69·57-s + 13.4·59-s − 5.75·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.839·5-s + 1.24·7-s + 0.333·9-s + 1.23·13-s − 0.484·15-s + 1.63·17-s − 0.388·19-s + 0.716·21-s + 0.353·23-s − 0.294·25-s + 0.192·27-s + 1.81·29-s − 0.285·31-s − 1.04·35-s + 0.746·37-s + 0.710·39-s + 0.521·41-s − 1.09·43-s − 0.279·45-s − 0.652·47-s + 0.539·49-s + 0.941·51-s − 0.0253·53-s − 0.224·57-s + 1.75·59-s − 0.736·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.114244850\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.114244850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 - 3.28T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 - 9.75T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 - 3.33T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 + 0.184T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 - 8.45T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.129264334531284823372041251469, −7.38532530552715210916963883516, −6.54266192841143731547907270407, −5.69553977292108452936618179775, −4.89660755379957055463950842754, −4.20463452505005421532302586334, −3.55492863029438337318198097578, −2.80804388058395370196172593860, −1.63085657376649837564411545998, −0.937569289000548086375443105324,
0.937569289000548086375443105324, 1.63085657376649837564411545998, 2.80804388058395370196172593860, 3.55492863029438337318198097578, 4.20463452505005421532302586334, 4.89660755379957055463950842754, 5.69553977292108452936618179775, 6.54266192841143731547907270407, 7.38532530552715210916963883516, 8.129264334531284823372041251469