| L(s) = 1 | + 1.93·3-s + 5-s − 4.93·7-s + 0.745·9-s − 0.745·11-s + 1.74·13-s + 1.93·15-s − 6.10·17-s − 5.44·19-s − 9.55·21-s + 23-s + 25-s − 4.36·27-s − 1.66·29-s + 1.61·31-s − 1.44·33-s − 4.93·35-s + 4.34·37-s + 3.37·39-s − 6.95·41-s − 5.01·43-s + 0.745·45-s − 2.68·47-s + 17.3·49-s − 11.8·51-s + 13.7·53-s − 0.745·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.447·5-s − 1.86·7-s + 0.248·9-s − 0.224·11-s + 0.484·13-s + 0.499·15-s − 1.48·17-s − 1.24·19-s − 2.08·21-s + 0.208·23-s + 0.200·25-s − 0.839·27-s − 0.309·29-s + 0.290·31-s − 0.251·33-s − 0.834·35-s + 0.714·37-s + 0.541·39-s − 1.08·41-s − 0.764·43-s + 0.111·45-s − 0.391·47-s + 2.47·49-s − 1.65·51-s + 1.88·53-s − 0.100·55-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 | \( 1 - 1.93T + 3T^{2} \) |
| 7 | \( 1 + 4.93T + 7T^{2} \) |
| 11 | \( 1 + 0.745T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 6.10T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 - 1.61T + 31T^{2} \) |
| 37 | \( 1 - 4.34T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 0.637T + 83T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−8.948108373543406266494868335251, −8.357112110892958758125044236474, −7.21269627305658072535104211342, −6.43814357190974763070888956997, −5.95354596325616201283000427203, −4.51143751555279354680828002521, −3.55748738603463917406044176550, −2.84505764976791200232834044772, −2.04127861578153616283218643204, 0,
2.04127861578153616283218643204, 2.84505764976791200232834044772, 3.55748738603463917406044176550, 4.51143751555279354680828002521, 5.95354596325616201283000427203, 6.43814357190974763070888956997, 7.21269627305658072535104211342, 8.357112110892958758125044236474, 8.948108373543406266494868335251
