| L(s) = 1 | − 3·3-s + 13.1·5-s + 15.6·7-s + 9·9-s + 55.7·11-s − 13·13-s − 39.3·15-s + 23.4·17-s + 25.6·19-s − 46.8·21-s − 189.·23-s + 47.2·25-s − 27·27-s + 236.·29-s + 47.0·31-s − 167.·33-s + 204.·35-s + 154.·37-s + 39·39-s − 34.6·41-s + 398.·43-s + 118.·45-s + 582.·47-s − 99.1·49-s − 70.4·51-s + 361.·53-s + 731.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.17·5-s + 0.843·7-s + 0.333·9-s + 1.52·11-s − 0.277·13-s − 0.677·15-s + 0.334·17-s + 0.309·19-s − 0.486·21-s − 1.71·23-s + 0.377·25-s − 0.192·27-s + 1.51·29-s + 0.272·31-s − 0.881·33-s + 0.989·35-s + 0.687·37-s + 0.160·39-s − 0.132·41-s + 1.41·43-s + 0.391·45-s + 1.80·47-s − 0.289·49-s − 0.193·51-s + 0.935·53-s + 1.79·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.262893889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.262893889\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 13.1T + 125T^{2} \) |
| 7 | \( 1 - 15.6T + 343T^{2} \) |
| 11 | \( 1 - 55.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 23.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 34.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 396.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 211.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 85.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 651.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 927.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 391.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 745.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 173.T + 9.12e5T^{2} \) |
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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
−8.663040339537159996800988353922, −7.77389835717044415361569491672, −6.88645624154527224327976326832, −6.04468920200788309190695031104, −5.70595912021884655323917946443, −4.61273054487641761704473496898, −3.99933366716651493691274307760, −2.53772379526104580160829602345, −1.62939953893538085647119128250, −0.893979176437509802330674656599,
0.893979176437509802330674656599, 1.62939953893538085647119128250, 2.53772379526104580160829602345, 3.99933366716651493691274307760, 4.61273054487641761704473496898, 5.70595912021884655323917946443, 6.04468920200788309190695031104, 6.88645624154527224327976326832, 7.77389835717044415361569491672, 8.663040339537159996800988353922
