| L(s) = 1 | − 3-s + 1.23·5-s + 3.23·7-s + 9-s + 11-s − 5.23·13-s − 1.23·15-s − 4.47·17-s − 6.47·19-s − 3.23·21-s − 5.70·23-s − 3.47·25-s − 27-s − 8.47·29-s − 33-s + 4.00·35-s − 6·37-s + 5.23·39-s + 10.9·41-s + 12.9·43-s + 1.23·45-s + 5.70·47-s + 3.47·49-s + 4.47·51-s − 6.76·53-s + 1.23·55-s + 6.47·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.552·5-s + 1.22·7-s + 0.333·9-s + 0.301·11-s − 1.45·13-s − 0.319·15-s − 1.08·17-s − 1.48·19-s − 0.706·21-s − 1.19·23-s − 0.694·25-s − 0.192·27-s − 1.57·29-s − 0.174·33-s + 0.676·35-s − 0.986·37-s + 0.838·39-s + 1.70·41-s + 1.97·43-s + 0.184·45-s + 0.832·47-s + 0.496·49-s + 0.626·51-s − 0.929·53-s + 0.166·55-s + 0.857·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 - 5.52T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 5.70T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−8.832770488508316862092390914030, −7.75534487076204499313172019845, −7.27153336175278825842120575829, −6.13852470612101959496263805352, −5.63945670483785377329613695233, −4.54106150368917466410470011820, −4.19965993873071508745118313619, −2.31023402744865386498102289771, −1.81578151088010343381606763798, 0,
1.81578151088010343381606763798, 2.31023402744865386498102289771, 4.19965993873071508745118313619, 4.54106150368917466410470011820, 5.63945670483785377329613695233, 6.13852470612101959496263805352, 7.27153336175278825842120575829, 7.75534487076204499313172019845, 8.832770488508316862092390914030
