| L(s) = 1 | + 3.17·5-s + 0.539·7-s − 1.82·11-s − 4.87·13-s − 0.617·17-s − 8.29·19-s − 4.09·23-s + 5.04·25-s + 8.66·29-s + 3.44·31-s + 1.70·35-s + 8.12·37-s + 41-s − 4.26·43-s − 2.61·47-s − 6.70·49-s − 4.97·53-s − 5.80·55-s + 0.183·59-s − 6.60·61-s − 15.4·65-s − 0.894·67-s − 14.9·71-s − 10.4·73-s − 0.986·77-s + 7.60·79-s + 13.6·83-s + ⋯ |
| L(s) = 1 | + 1.41·5-s + 0.203·7-s − 0.551·11-s − 1.35·13-s − 0.149·17-s − 1.90·19-s − 0.853·23-s + 1.00·25-s + 1.60·29-s + 0.619·31-s + 0.288·35-s + 1.33·37-s + 0.156·41-s − 0.649·43-s − 0.381·47-s − 0.958·49-s − 0.682·53-s − 0.782·55-s + 0.0238·59-s − 0.845·61-s − 1.91·65-s − 0.109·67-s − 1.77·71-s − 1.21·73-s − 0.112·77-s + 0.855·79-s + 1.50·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 - 0.539T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 + 0.617T + 17T^{2} \) |
| 19 | \( 1 + 8.29T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 8.66T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 8.12T + 37T^{2} \) |
| 43 | \( 1 + 4.26T + 43T^{2} \) |
| 47 | \( 1 + 2.61T + 47T^{2} \) |
| 53 | \( 1 + 4.97T + 53T^{2} \) |
| 59 | \( 1 - 0.183T + 59T^{2} \) |
| 61 | \( 1 + 6.60T + 61T^{2} \) |
| 67 | \( 1 + 0.894T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 7.60T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 4.15T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.935616346207849328791444315547, −6.79853733224971367510752424950, −6.32952291800851219842389121461, −5.67929012836439786473160886006, −4.75652659340300641133111845076, −4.39972653480907626237721716553, −2.84143565278835135905710249789, −2.37236848140754011962657673040, −1.56126685567654665656631777956, 0,
1.56126685567654665656631777956, 2.37236848140754011962657673040, 2.84143565278835135905710249789, 4.39972653480907626237721716553, 4.75652659340300641133111845076, 5.67929012836439786473160886006, 6.32952291800851219842389121461, 6.79853733224971367510752424950, 7.935616346207849328791444315547
