| L(s) = 1 | − 5-s + 1.44·7-s + 2.44·11-s − 0.449·13-s + 0.550·17-s − 6.44·19-s + 23-s + 25-s − 7.89·29-s − 7·31-s − 1.44·35-s − 8.34·37-s − 1.89·41-s + 0.898·43-s + 2.44·47-s − 4.89·49-s + 10.3·53-s − 2.44·55-s + 7.89·59-s + 9.34·61-s + 0.449·65-s − 3.44·67-s + 3·71-s − 5.34·73-s + 3.55·77-s − 4·79-s + 4.34·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.547·7-s + 0.738·11-s − 0.124·13-s + 0.133·17-s − 1.47·19-s + 0.208·23-s + 0.200·25-s − 1.46·29-s − 1.25·31-s − 0.245·35-s − 1.37·37-s − 0.296·41-s + 0.137·43-s + 0.357·47-s − 0.699·49-s + 1.42·53-s − 0.330·55-s + 1.02·59-s + 1.19·61-s + 0.0557·65-s − 0.421·67-s + 0.356·71-s − 0.625·73-s + 0.404·77-s − 0.450·79-s + 0.477·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 - 0.550T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 8.34T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 - 0.898T + 43T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 7.89T + 59T^{2} \) |
| 61 | \( 1 - 9.34T + 61T^{2} \) |
| 67 | \( 1 + 3.44T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 5.34T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 + 5.10T + 97T^{2} \) |
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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
−8.143455600495389688142905588624, −7.22551589649175179735293745402, −6.78852547306565040913174168804, −5.75101697185042016350128208012, −5.08424236614450145806971150297, −4.06756244057579983824009411587, −3.66958406888274022395988118573, −2.33778197045862962078106250639, −1.48352208041035441193695489875, 0,
1.48352208041035441193695489875, 2.33778197045862962078106250639, 3.66958406888274022395988118573, 4.06756244057579983824009411587, 5.08424236614450145806971150297, 5.75101697185042016350128208012, 6.78852547306565040913174168804, 7.22551589649175179735293745402, 8.143455600495389688142905588624
