| L(s) = 1 | + 8.30·3-s − 11.7·5-s − 23.1·7-s + 42.0·9-s + 62.0·11-s + 55.0·13-s − 97.3·15-s + 23.4·17-s − 36.9·19-s − 192.·21-s − 28.7·23-s + 12.2·25-s + 124.·27-s + 29·29-s − 231.·31-s + 515.·33-s + 271.·35-s + 202.·37-s + 456.·39-s − 253.·41-s − 127.·43-s − 492.·45-s + 462.·47-s + 195.·49-s + 194.·51-s + 632.·53-s − 727.·55-s + ⋯ |
| L(s) = 1 | + 1.59·3-s − 1.04·5-s − 1.25·7-s + 1.55·9-s + 1.70·11-s + 1.17·13-s − 1.67·15-s + 0.333·17-s − 0.446·19-s − 2.00·21-s − 0.260·23-s + 0.0981·25-s + 0.889·27-s + 0.185·29-s − 1.34·31-s + 2.72·33-s + 1.31·35-s + 0.898·37-s + 1.87·39-s − 0.964·41-s − 0.451·43-s − 1.63·45-s + 1.43·47-s + 0.569·49-s + 0.533·51-s + 1.63·53-s − 1.78·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.286106827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.286106827\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 8.30T + 27T^{2} \) |
| 5 | \( 1 + 11.7T + 125T^{2} \) |
| 7 | \( 1 + 23.1T + 343T^{2} \) |
| 11 | \( 1 - 62.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.7T + 1.21e4T^{2} \) |
| 31 | \( 1 + 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 462.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 632.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 735.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 385.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 63.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 238.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 658.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 916.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.07e3T + 9.12e5T^{2} \) |
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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
−8.739503186927812630145143221217, −8.378441207587690475236227478701, −7.34340070067565779498741329991, −6.77446611070128451338289023133, −5.89027819892971490130155885627, −4.09349471705012160864093516678, −3.77565438335665866912337577269, −3.26334369207011048644302910978, −2.02425812865306084426247255718, −0.793418299189879051110586311796,
0.793418299189879051110586311796, 2.02425812865306084426247255718, 3.26334369207011048644302910978, 3.77565438335665866912337577269, 4.09349471705012160864093516678, 5.89027819892971490130155885627, 6.77446611070128451338289023133, 7.34340070067565779498741329991, 8.378441207587690475236227478701, 8.739503186927812630145143221217
