| L(s) = 1 | − 5-s + 4.17·7-s + 5.52·11-s + 13-s + 0.703·17-s + 6.82·19-s + 2.64·23-s + 25-s − 8.17·29-s − 9.52·31-s − 4.17·35-s − 6.87·37-s − 0.703·41-s − 1.35·43-s + 8.17·47-s + 10.4·49-s − 5.04·53-s − 5.52·55-s + 12.2·59-s − 0.172·61-s − 65-s + 10.8·67-s + 5.52·71-s − 11.2·73-s + 23.0·77-s + 1.29·79-s + 9.58·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.57·7-s + 1.66·11-s + 0.277·13-s + 0.170·17-s + 1.56·19-s + 0.552·23-s + 0.200·25-s − 1.51·29-s − 1.71·31-s − 0.705·35-s − 1.13·37-s − 0.109·41-s − 0.206·43-s + 1.19·47-s + 1.48·49-s − 0.693·53-s − 0.744·55-s + 1.59·59-s − 0.0220·61-s − 0.124·65-s + 1.32·67-s + 0.655·71-s − 1.31·73-s + 2.62·77-s + 0.145·79-s + 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.352873817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.352873817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4.17T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 17 | \( 1 - 0.703T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 + 9.52T + 31T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 + 0.703T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 8.17T + 47T^{2} \) |
| 53 | \( 1 + 5.04T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 0.172T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 1.29T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−9.008743273665653353581885371984, −8.222475729931087775448286941630, −7.39700467908422568410746712565, −6.94830021082220756603381757969, −5.63272866520891903977752982745, −5.10914575956725825283776330417, −4.02221510137209967852801267034, −3.49324100677638905982521516143, −1.88028873824518586110939663863, −1.11690548439760951688231015451,
1.11690548439760951688231015451, 1.88028873824518586110939663863, 3.49324100677638905982521516143, 4.02221510137209967852801267034, 5.10914575956725825283776330417, 5.63272866520891903977752982745, 6.94830021082220756603381757969, 7.39700467908422568410746712565, 8.222475729931087775448286941630, 9.008743273665653353581885371984
