| L(s) = 1 | + 7-s − 3·9-s − 2·11-s + 2·13-s − 3·17-s + 2·19-s + 23-s + 7·29-s + 5·31-s − 11·37-s + 41-s − 6·49-s − 11·53-s + 13·59-s − 8·61-s − 3·63-s + 5·67-s − 5·71-s − 6·73-s − 2·77-s + 12·79-s + 9·81-s + 9·83-s + 4·89-s + 2·91-s + 14·97-s + 6·99-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.603·11-s + 0.554·13-s − 0.727·17-s + 0.458·19-s + 0.208·23-s + 1.29·29-s + 0.898·31-s − 1.80·37-s + 0.156·41-s − 6/7·49-s − 1.51·53-s + 1.69·59-s − 1.02·61-s − 0.377·63-s + 0.610·67-s − 0.593·71-s − 0.702·73-s − 0.227·77-s + 1.35·79-s + 81-s + 0.987·83-s + 0.423·89-s + 0.209·91-s + 1.42·97-s + 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.44578516743869049752414932792, −6.56204616709562076476583837744, −6.12098531810167623331505780632, −5.11210839347486606415315969228, −4.86746669962990741149290628541, −3.74098577934752174502652790143, −3.00081649567865073512470760829, −2.29431465404175593878605173379, −1.20850711303608592687633946299, 0,
1.20850711303608592687633946299, 2.29431465404175593878605173379, 3.00081649567865073512470760829, 3.74098577934752174502652790143, 4.86746669962990741149290628541, 5.11210839347486606415315969228, 6.12098531810167623331505780632, 6.56204616709562076476583837744, 7.44578516743869049752414932792
