| L(s) = 1 | + 2-s + 3-s + 4-s − 2.33·5-s + 6-s + 4.77·7-s + 8-s + 9-s − 2.33·10-s + 2.17·11-s + 12-s − 6.38·13-s + 4.77·14-s − 2.33·15-s + 16-s + 5.62·17-s + 18-s − 2.33·20-s + 4.77·21-s + 2.17·22-s + 4·23-s + 24-s + 0.433·25-s − 6.38·26-s + 27-s + 4.77·28-s − 1.14·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.04·5-s + 0.408·6-s + 1.80·7-s + 0.353·8-s + 0.333·9-s − 0.737·10-s + 0.656·11-s + 0.288·12-s − 1.77·13-s + 1.27·14-s − 0.601·15-s + 0.250·16-s + 1.36·17-s + 0.235·18-s − 0.521·20-s + 1.04·21-s + 0.464·22-s + 0.834·23-s + 0.204·24-s + 0.0866·25-s − 1.25·26-s + 0.192·27-s + 0.901·28-s − 0.211·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.623823529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.623823529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2.33T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 17 | \( 1 - 5.62T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 7.67T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 5.89T + 47T^{2} \) |
| 53 | \( 1 - 6.51T + 53T^{2} \) |
| 59 | \( 1 + 4.29T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 + 7.42T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 - 4.86T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 7.91T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.878850419750957877529049376534, −8.016016167652822974105986251237, −7.57306313687490511972807823492, −7.07542955518748034111123466325, −5.63502562438727356363174288667, −4.77594131693896220080811447361, −4.36346203394076162931086089661, −3.35968117639323075302363622398, −2.35966939166870005609012002947, −1.21845743071698662979001414055,
1.21845743071698662979001414055, 2.35966939166870005609012002947, 3.35968117639323075302363622398, 4.36346203394076162931086089661, 4.77594131693896220080811447361, 5.63502562438727356363174288667, 7.07542955518748034111123466325, 7.57306313687490511972807823492, 8.016016167652822974105986251237, 8.878850419750957877529049376534
