| L(s) = 1 | − 1.23·2-s + 3·3-s − 6.46·4-s + 5·5-s − 3.71·6-s − 14.9·7-s + 17.9·8-s + 9·9-s − 6.18·10-s + 35.3·11-s − 19.4·12-s − 88.9·13-s + 18.5·14-s + 15·15-s + 29.5·16-s + 58.2·17-s − 11.1·18-s + 110.·19-s − 32.3·20-s − 44.8·21-s − 43.6·22-s + 179.·23-s + 53.7·24-s + 25·25-s + 110.·26-s + 27·27-s + 96.7·28-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.808·4-s + 0.447·5-s − 0.252·6-s − 0.807·7-s + 0.791·8-s + 0.333·9-s − 0.195·10-s + 0.967·11-s − 0.466·12-s − 1.89·13-s + 0.353·14-s + 0.258·15-s + 0.462·16-s + 0.831·17-s − 0.145·18-s + 1.32·19-s − 0.361·20-s − 0.466·21-s − 0.423·22-s + 1.62·23-s + 0.456·24-s + 0.200·25-s + 0.830·26-s + 0.192·27-s + 0.652·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.777490736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777490736\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 163 | \( 1 + 163T \) |
| good | 2 | \( 1 + 1.23T + 8T^{2} \) |
| 7 | \( 1 + 14.9T + 343T^{2} \) |
| 11 | \( 1 - 35.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 98.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 4.88T + 5.06e4T^{2} \) |
| 41 | \( 1 + 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 423.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 604.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 775.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 564.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 571.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 798.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 107.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 824.T + 9.12e5T^{2} \) |
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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
−8.745011095385537099560051505725, −7.944057291423290172647220309527, −7.12409908863559179514811596399, −6.54943980743948010557768080763, −5.11141038105890807844690198370, −4.86971903956321363918312678317, −3.47798735966974353872090154478, −2.97241709581435360033769761133, −1.59887984935814121278651528167, −0.64595124355527721367525580156,
0.64595124355527721367525580156, 1.59887984935814121278651528167, 2.97241709581435360033769761133, 3.47798735966974353872090154478, 4.86971903956321363918312678317, 5.11141038105890807844690198370, 6.54943980743948010557768080763, 7.12409908863559179514811596399, 7.944057291423290172647220309527, 8.745011095385537099560051505725
