| L(s) = 1 | + 25.2·3-s + 55.5·5-s + 200.·7-s + 395.·9-s + 121·11-s − 727.·13-s + 1.40e3·15-s − 1.10e3·17-s − 1.66e3·19-s + 5.07e3·21-s + 2.98e3·23-s − 40.0·25-s + 3.85e3·27-s + 1.55e3·29-s + 9.51e3·31-s + 3.05e3·33-s + 1.11e4·35-s + 9.43e3·37-s − 1.83e4·39-s + 7.37e3·41-s + 8.52e3·43-s + 2.19e4·45-s + 3.00e4·47-s + 2.35e4·49-s − 2.80e4·51-s − 2.39e4·53-s + 6.72e3·55-s + ⋯ |
| L(s) = 1 | + 1.62·3-s + 0.993·5-s + 1.54·7-s + 1.62·9-s + 0.301·11-s − 1.19·13-s + 1.61·15-s − 0.930·17-s − 1.05·19-s + 2.51·21-s + 1.17·23-s − 0.0128·25-s + 1.01·27-s + 0.342·29-s + 1.77·31-s + 0.488·33-s + 1.53·35-s + 1.13·37-s − 1.93·39-s + 0.684·41-s + 0.703·43-s + 1.61·45-s + 1.98·47-s + 1.39·49-s − 1.50·51-s − 1.17·53-s + 0.299·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.307981162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.307981162\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 - 25.2T + 243T^{2} \) |
| 5 | \( 1 - 55.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 200.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 727.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.96e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.90e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.88e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.37e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.91e3T + 8.58e9T^{2} \) |
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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
−9.389813301140849158726246338354, −8.881217079265874155313461584433, −8.057147655590802014277155121830, −7.36666750851791548502486630243, −6.23172673601397853986420892717, −4.82460565862799976683098065074, −4.27906679631701291060093210354, −2.58249390203976490695454581090, −2.26239687125334516244887884937, −1.19629078210578017810573943444,
1.19629078210578017810573943444, 2.26239687125334516244887884937, 2.58249390203976490695454581090, 4.27906679631701291060093210354, 4.82460565862799976683098065074, 6.23172673601397853986420892717, 7.36666750851791548502486630243, 8.057147655590802014277155121830, 8.881217079265874155313461584433, 9.389813301140849158726246338354
