| L(s) = 1 | − 2.18·2-s − 6.08·3-s − 3.21·4-s − 5·5-s + 13.3·6-s − 10.8·7-s + 24.5·8-s + 10.0·9-s + 10.9·10-s − 0.656·11-s + 19.5·12-s + 48.6·13-s + 23.7·14-s + 30.4·15-s − 27.8·16-s − 21.9·18-s + 118.·19-s + 16.0·20-s + 65.9·21-s + 1.43·22-s − 84.7·23-s − 149.·24-s + 25·25-s − 106.·26-s + 103.·27-s + 34.8·28-s − 84.8·29-s + ⋯ |
| L(s) = 1 | − 0.773·2-s − 1.17·3-s − 0.402·4-s − 0.447·5-s + 0.905·6-s − 0.585·7-s + 1.08·8-s + 0.372·9-s + 0.345·10-s − 0.0180·11-s + 0.471·12-s + 1.03·13-s + 0.452·14-s + 0.523·15-s − 0.435·16-s − 0.287·18-s + 1.43·19-s + 0.179·20-s + 0.685·21-s + 0.0139·22-s − 0.768·23-s − 1.27·24-s + 0.200·25-s − 0.802·26-s + 0.735·27-s + 0.235·28-s − 0.543·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4819606934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4819606934\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.18T + 8T^{2} \) |
| 3 | \( 1 + 6.08T + 27T^{2} \) |
| 7 | \( 1 + 10.8T + 343T^{2} \) |
| 11 | \( 1 + 0.656T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.6T + 2.19e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 84.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 32.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 90.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 46.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 385.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 354.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 105.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 330.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 243.T + 9.12e5T^{2} \) |
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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
−9.264015299472788636266028092441, −8.330102542865442599067637749740, −7.71504688612725897941508617068, −6.66446708933174166141748180258, −5.95144125666893845827443011034, −5.06078503781473533336726842686, −4.17991747171116872897776538920, −3.13842779228542787404779102128, −1.29905251359393406751300336094, −0.47214166205124654929816788079,
0.47214166205124654929816788079, 1.29905251359393406751300336094, 3.13842779228542787404779102128, 4.17991747171116872897776538920, 5.06078503781473533336726842686, 5.95144125666893845827443011034, 6.66446708933174166141748180258, 7.71504688612725897941508617068, 8.330102542865442599067637749740, 9.264015299472788636266028092441
