| L(s) = 1 | − 2.82·3-s + 5-s + 2·7-s + 5.00·9-s + 11-s + 1.17·13-s − 2.82·15-s + 6.82·17-s − 5.65·21-s + 2.82·23-s + 25-s − 5.65·27-s + 3.65·29-s − 2.82·33-s + 2·35-s + 7.65·37-s − 3.31·39-s + 6·41-s − 6·43-s + 5.00·45-s − 2.82·47-s − 3·49-s − 19.3·51-s − 11.6·53-s + 55-s + 1.65·59-s + 9.31·61-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.447·5-s + 0.755·7-s + 1.66·9-s + 0.301·11-s + 0.324·13-s − 0.730·15-s + 1.65·17-s − 1.23·21-s + 0.589·23-s + 0.200·25-s − 1.08·27-s + 0.679·29-s − 0.492·33-s + 0.338·35-s + 1.25·37-s − 0.530·39-s + 0.937·41-s − 0.914·43-s + 0.745·45-s − 0.412·47-s − 0.428·49-s − 2.70·51-s − 1.60·53-s + 0.134·55-s + 0.215·59-s + 1.19·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477724323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477724323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.463130382661198694286618949489, −7.74398804559125571126744553631, −6.87734370983067905926432292651, −6.18826981559994314762283502177, −5.56413631026318692494339041461, −4.98561503530335547366849495895, −4.24499997232995231466472873806, −3.04151447545167000495121765655, −1.56541718617463956231940835708, −0.867643422044574825094590118017,
0.867643422044574825094590118017, 1.56541718617463956231940835708, 3.04151447545167000495121765655, 4.24499997232995231466472873806, 4.98561503530335547366849495895, 5.56413631026318692494339041461, 6.18826981559994314762283502177, 6.87734370983067905926432292651, 7.74398804559125571126744553631, 8.463130382661198694286618949489
