L(s) = 1 | + 0.289·2-s − 3-s − 1.91·4-s + 5-s − 0.289·6-s − 4.91·7-s − 1.13·8-s + 9-s + 0.289·10-s − 4.91·11-s + 1.91·12-s − 1.42·14-s − 15-s + 3.50·16-s − 4.33·17-s + 0.289·18-s − 2.57·19-s − 1.91·20-s + 4.91·21-s − 1.42·22-s − 6.33·23-s + 1.13·24-s + 25-s − 27-s + 9.42·28-s + 6·29-s − 0.289·30-s + ⋯ |
L(s) = 1 | + 0.204·2-s − 0.577·3-s − 0.958·4-s + 0.447·5-s − 0.118·6-s − 1.85·7-s − 0.400·8-s + 0.333·9-s + 0.0914·10-s − 1.48·11-s + 0.553·12-s − 0.379·14-s − 0.258·15-s + 0.876·16-s − 1.05·17-s + 0.0681·18-s − 0.591·19-s − 0.428·20-s + 1.07·21-s − 0.303·22-s − 1.32·23-s + 0.231·24-s + 0.200·25-s − 0.192·27-s + 1.78·28-s + 1.11·29-s − 0.0527·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2963215184\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2963215184\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.289T + 2T^{2} \) |
| 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 + 9.49T + 37T^{2} \) |
| 41 | \( 1 + 4.33T + 41T^{2} \) |
| 43 | \( 1 + 1.15T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 + 0.338T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 + 0.916T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 0.338T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.926430687702851024944354309520, −8.350697896369600912359735237964, −7.16771551163394357574782202742, −6.39881037045957818658753095484, −5.80380255339942910751684899717, −5.06619564626128607776603266132, −4.16362712542994981314041242266, −3.26604638100967708019004254494, −2.29350174652220423653592508986, −0.32120513334294965096213094632,
0.32120513334294965096213094632, 2.29350174652220423653592508986, 3.26604638100967708019004254494, 4.16362712542994981314041242266, 5.06619564626128607776603266132, 5.80380255339942910751684899717, 6.39881037045957818658753095484, 7.16771551163394357574782202742, 8.350697896369600912359735237964, 8.926430687702851024944354309520