| L(s) = 1 | − 3·9-s + 1.30·11-s + 6.93·23-s − 5·25-s − 10.7·29-s + 9.99·37-s + 11.6·43-s − 14.5·53-s − 14.5·67-s − 5.83·71-s − 17.3·79-s + 9·81-s − 3.92·99-s + 0.708·107-s + 2.94·109-s + 7.38·113-s + ⋯ |
| L(s) = 1 | − 9-s + 0.394·11-s + 1.44·23-s − 25-s − 1.99·29-s + 1.64·37-s + 1.77·43-s − 1.99·53-s − 1.78·67-s − 0.692·71-s − 1.95·79-s + 81-s − 0.394·99-s + 0.0684·107-s + 0.282·109-s + 0.694·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 + 10.7T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9.99T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.66336459789909335955007596398, −7.29999222736340672477013964196, −6.08443581573594851645427885410, −5.87740185563301427864362241295, −4.89680777858398397519361876420, −4.07453562232766656273925261945, −3.21568087348060776020630705690, −2.45168396550661001473369279307, −1.33048686812061479401579363064, 0,
1.33048686812061479401579363064, 2.45168396550661001473369279307, 3.21568087348060776020630705690, 4.07453562232766656273925261945, 4.89680777858398397519361876420, 5.87740185563301427864362241295, 6.08443581573594851645427885410, 7.29999222736340672477013964196, 7.66336459789909335955007596398
