| L(s) = 1 | − 3.74·2-s − 3·3-s + 6.03·4-s + 21.0·5-s + 11.2·6-s + 27.2·7-s + 7.34·8-s + 9·9-s − 78.8·10-s − 43.3·11-s − 18.1·12-s − 19.7·13-s − 102.·14-s − 63.1·15-s − 75.8·16-s + 56.8·17-s − 33.7·18-s + 127.·20-s − 81.8·21-s + 162.·22-s − 10.6·23-s − 22.0·24-s + 317.·25-s + 74.1·26-s − 27·27-s + 164.·28-s − 47.7·29-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 0.577·3-s + 0.754·4-s + 1.88·5-s + 0.764·6-s + 1.47·7-s + 0.324·8-s + 0.333·9-s − 2.49·10-s − 1.18·11-s − 0.435·12-s − 0.422·13-s − 1.95·14-s − 1.08·15-s − 1.18·16-s + 0.810·17-s − 0.441·18-s + 1.42·20-s − 0.850·21-s + 1.57·22-s − 0.0962·23-s − 0.187·24-s + 2.54·25-s + 0.559·26-s − 0.192·27-s + 1.11·28-s − 0.305·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.448070459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.448070459\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 3.74T + 8T^{2} \) |
| 5 | \( 1 - 21.0T + 125T^{2} \) |
| 7 | \( 1 - 27.2T + 343T^{2} \) |
| 11 | \( 1 + 43.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.8T + 4.91e3T^{2} \) |
| 23 | \( 1 + 10.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 47.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 368.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 421.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 725.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 161.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 417.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 462.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 197.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 7.82T + 5.71e5T^{2} \) |
| 89 | \( 1 - 559.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 743.T + 9.12e5T^{2} \) |
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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
−9.668591813525619952746139498699, −8.777405602038501372139447584798, −7.928495699658649065873109034405, −7.27979711170714688136780112169, −6.07739858386420426148924297465, −5.28065464191846009205006411573, −4.72433988912116840203400903880, −2.47147133518128759502927792045, −1.74002992876192515767308366426, −0.842515777875383586052001794393,
0.842515777875383586052001794393, 1.74002992876192515767308366426, 2.47147133518128759502927792045, 4.72433988912116840203400903880, 5.28065464191846009205006411573, 6.07739858386420426148924297465, 7.27979711170714688136780112169, 7.928495699658649065873109034405, 8.777405602038501372139447584798, 9.668591813525619952746139498699
