| L(s) = 1 | − 3.29·3-s + 0.224·5-s + 1.38·7-s + 7.82·9-s + 11-s + 4.56·13-s − 0.738·15-s + 6.79·17-s − 5.08·19-s − 4.56·21-s − 2.53·23-s − 4.94·25-s − 15.8·27-s + 1.59·29-s − 10.8·31-s − 3.29·33-s + 0.311·35-s + 0.774·37-s − 15.0·39-s − 5.97·41-s + 0.900·43-s + 1.75·45-s − 1.69·47-s − 5.07·49-s − 22.3·51-s − 0.188·53-s + 0.224·55-s + ⋯ |
| L(s) = 1 | − 1.89·3-s + 0.100·5-s + 0.523·7-s + 2.60·9-s + 0.301·11-s + 1.26·13-s − 0.190·15-s + 1.64·17-s − 1.16·19-s − 0.995·21-s − 0.528·23-s − 0.989·25-s − 3.05·27-s + 0.295·29-s − 1.95·31-s − 0.572·33-s + 0.0526·35-s + 0.127·37-s − 2.40·39-s − 0.933·41-s + 0.137·43-s + 0.262·45-s − 0.246·47-s − 0.725·49-s − 3.12·51-s − 0.0258·53-s + 0.0302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + 3.29T + 3T^{2} \) |
| 5 | \( 1 - 0.224T + 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 0.774T + 37T^{2} \) |
| 41 | \( 1 + 5.97T + 41T^{2} \) |
| 43 | \( 1 - 0.900T + 43T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 + 0.188T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 + 0.834T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 0.605T + 71T^{2} \) |
| 73 | \( 1 + 2.02T + 73T^{2} \) |
| 79 | \( 1 - 8.16T + 79T^{2} \) |
| 83 | \( 1 + 5.40T + 83T^{2} \) |
| 89 | \( 1 - 6.67T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
| 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
−7.65224384938220479015396011816, −6.73125410405146910896176648085, −6.19866997980535321972925913926, −5.59422050556470657945708602403, −5.11781150753135818236739572947, −4.11138372932739390788435487371, −3.63188465937611471141020890890, −1.82139964560032085835085957822, −1.23202812135306455910119323356, 0,
1.23202812135306455910119323356, 1.82139964560032085835085957822, 3.63188465937611471141020890890, 4.11138372932739390788435487371, 5.11781150753135818236739572947, 5.59422050556470657945708602403, 6.19866997980535321972925913926, 6.73125410405146910896176648085, 7.65224384938220479015396011816
