| L(s) = 1 | + 2.14·3-s − 5-s + 1.14·7-s + 1.60·9-s − 5.89·11-s − 4.89·13-s − 2.14·15-s − 5.89·17-s + 2.34·19-s + 2.45·21-s − 23-s + 25-s − 3.00·27-s + 3.74·29-s − 5.68·31-s − 12.6·33-s − 1.14·35-s + 4·37-s − 10.4·39-s − 1.05·41-s − 11.4·43-s − 1.60·45-s − 7.74·47-s − 5.68·49-s − 12.6·51-s + 12.9·53-s + 5.89·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 0.447·5-s + 0.432·7-s + 0.533·9-s − 1.77·11-s − 1.35·13-s − 0.553·15-s − 1.42·17-s + 0.538·19-s + 0.536·21-s − 0.208·23-s + 0.200·25-s − 0.577·27-s + 0.695·29-s − 1.02·31-s − 2.20·33-s − 0.193·35-s + 0.657·37-s − 1.68·39-s − 0.165·41-s − 1.75·43-s − 0.238·45-s − 1.12·47-s − 0.812·49-s − 1.76·51-s + 1.78·53-s + 0.794·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 0.797T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 0.912T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
| 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
−8.584447899011735667339580925616, −8.172579301285071387617011357481, −7.51702056163198319132966030292, −6.81313382867175619352484404742, −5.29990444401821595532093900436, −4.78767073186704803329905978589, −3.65522218727165515436571081491, −2.64700778070461331394599302020, −2.14756264929705504930733200813, 0,
2.14756264929705504930733200813, 2.64700778070461331394599302020, 3.65522218727165515436571081491, 4.78767073186704803329905978589, 5.29990444401821595532093900436, 6.81313382867175619352484404742, 7.51702056163198319132966030292, 8.172579301285071387617011357481, 8.584447899011735667339580925616
