| L(s) = 1 | + 4.41·2-s + 7.84·3-s + 11.5·4-s − 5·5-s + 34.6·6-s − 8.97·7-s + 15.5·8-s + 34.4·9-s − 22.0·10-s − 28.9·11-s + 90.2·12-s + 16.0·13-s − 39.6·14-s − 39.2·15-s − 23.5·16-s + 25.1·17-s + 152.·18-s − 35.6·19-s − 57.5·20-s − 70.3·21-s − 127.·22-s − 23·23-s + 121.·24-s + 25·25-s + 71.0·26-s + 58.7·27-s − 103.·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.50·3-s + 1.43·4-s − 0.447·5-s + 2.35·6-s − 0.484·7-s + 0.685·8-s + 1.27·9-s − 0.698·10-s − 0.792·11-s + 2.17·12-s + 0.343·13-s − 0.756·14-s − 0.674·15-s − 0.367·16-s + 0.359·17-s + 1.99·18-s − 0.430·19-s − 0.643·20-s − 0.731·21-s − 1.23·22-s − 0.208·23-s + 1.03·24-s + 0.200·25-s + 0.536·26-s + 0.418·27-s − 0.697·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.732285736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.732285736\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 4.41T + 8T^{2} \) |
| 3 | \( 1 - 7.84T + 27T^{2} \) |
| 7 | \( 1 + 8.97T + 343T^{2} \) |
| 11 | \( 1 + 28.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 40.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 379.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 402.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 421.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 755.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 307.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 319.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 554.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 705.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 455.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−13.18529092100911363295159193701, −12.65693415274224062408424496969, −11.34535917479340958890934187129, −9.900379357342237298993292327145, −8.579730196168370893840188967713, −7.54906478306587026346711779162, −6.16502609749531657616849402475, −4.55900292119228544154524619392, −3.44421399713674782559826242602, −2.55568974584814226338514522095,
2.55568974584814226338514522095, 3.44421399713674782559826242602, 4.55900292119228544154524619392, 6.16502609749531657616849402475, 7.54906478306587026346711779162, 8.579730196168370893840188967713, 9.900379357342237298993292327145, 11.34535917479340958890934187129, 12.65693415274224062408424496969, 13.18529092100911363295159193701
