| L(s) = 1 | + 28·7-s + 24·11-s + 70·13-s + 102·17-s + 20·19-s − 72·23-s − 306·29-s − 136·31-s + 214·37-s + 150·41-s + 292·43-s − 72·47-s + 441·49-s − 414·53-s + 744·59-s − 418·61-s − 188·67-s − 480·71-s − 434·73-s + 672·77-s + 1.35e3·79-s − 612·83-s + 30·89-s + 1.96e3·91-s + 286·97-s + 1.54e3·101-s − 1.17e3·103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.657·11-s + 1.49·13-s + 1.45·17-s + 0.241·19-s − 0.652·23-s − 1.95·29-s − 0.787·31-s + 0.950·37-s + 0.571·41-s + 1.03·43-s − 0.223·47-s + 9/7·49-s − 1.07·53-s + 1.64·59-s − 0.877·61-s − 0.342·67-s − 0.802·71-s − 0.695·73-s + 0.994·77-s + 1.92·79-s − 0.809·83-s + 0.0357·89-s + 2.25·91-s + 0.299·97-s + 1.51·101-s − 1.12·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.021766060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.021766060\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 306 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 150 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 744 T + p^{3} T^{2} \) |
| 61 | \( 1 + 418 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 480 T + p^{3} T^{2} \) |
| 73 | \( 1 + 434 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
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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
−9.627107985506255368681197951507, −8.846522938993848500527252450157, −7.951745927688524586685551393291, −7.45635035216127968454771538411, −6.04522802948136907223445438717, −5.47506284608224696953921524591, −4.26530596992240795167773585319, −3.49424895476761048810351244596, −1.84689834420561360588125437004, −1.06507622651369448613257778298,
1.06507622651369448613257778298, 1.84689834420561360588125437004, 3.49424895476761048810351244596, 4.26530596992240795167773585319, 5.47506284608224696953921524591, 6.04522802948136907223445438717, 7.45635035216127968454771538411, 7.951745927688524586685551393291, 8.846522938993848500527252450157, 9.627107985506255368681197951507
