| L(s) = 1 | + 3-s + 5-s + 2.56·7-s + 9-s + 1.43·11-s + 13-s + 15-s + 5.68·17-s + 5.12·19-s + 2.56·21-s + 1.43·23-s + 25-s + 27-s − 2·29-s − 1.12·31-s + 1.43·33-s + 2.56·35-s − 10.8·37-s + 39-s − 9.68·41-s − 6.24·43-s + 45-s + 1.12·47-s − 0.438·49-s + 5.68·51-s − 0.561·53-s + 1.43·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.968·7-s + 0.333·9-s + 0.433·11-s + 0.277·13-s + 0.258·15-s + 1.37·17-s + 1.17·19-s + 0.558·21-s + 0.299·23-s + 0.200·25-s + 0.192·27-s − 0.371·29-s − 0.201·31-s + 0.250·33-s + 0.432·35-s − 1.77·37-s + 0.160·39-s − 1.51·41-s − 0.952·43-s + 0.149·45-s + 0.163·47-s − 0.0626·49-s + 0.796·51-s − 0.0771·53-s + 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.228191193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.228191193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 + 0.561T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 + 0.246T + 73T^{2} \) |
| 79 | \( 1 - 8.80T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−8.617290316311767517371589162387, −8.033633192173720220709723376374, −7.29846658512509333788672444589, −6.55608559380942687083707979100, −5.34235889094039776567565353095, −5.10141067262799425280761502729, −3.76657340189447150299955761631, −3.18313748762880687065243988842, −1.89801378018743872347424133607, −1.20918315853101573553955391569,
1.20918315853101573553955391569, 1.89801378018743872347424133607, 3.18313748762880687065243988842, 3.76657340189447150299955761631, 5.10141067262799425280761502729, 5.34235889094039776567565353095, 6.55608559380942687083707979100, 7.29846658512509333788672444589, 8.033633192173720220709723376374, 8.617290316311767517371589162387
