| L(s) = 1 | − 5-s + 4.93·7-s − 0.745·11-s + 1.74·13-s + 6.10·17-s + 5.44·19-s + 23-s + 25-s + 1.66·29-s − 1.61·31-s − 4.93·35-s + 4.34·37-s + 6.95·41-s + 5.01·43-s − 2.68·47-s + 17.3·49-s − 13.7·53-s + 0.745·55-s − 12.2·59-s − 13.9·61-s − 1.74·65-s + 13.1·67-s + 9.67·71-s + 5.69·73-s − 3.68·77-s + 10.3·79-s − 0.637·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.86·7-s − 0.224·11-s + 0.484·13-s + 1.48·17-s + 1.24·19-s + 0.208·23-s + 0.200·25-s + 0.309·29-s − 0.290·31-s − 0.834·35-s + 0.714·37-s + 1.08·41-s + 0.764·43-s − 0.391·47-s + 2.47·49-s − 1.88·53-s + 0.100·55-s − 1.59·59-s − 1.78·61-s − 0.216·65-s + 1.60·67-s + 1.14·71-s + 0.666·73-s − 0.419·77-s + 1.16·79-s − 0.0699·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.935085983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.935085983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4.93T + 7T^{2} \) |
| 11 | \( 1 + 0.745T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 - 6.10T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 29 | \( 1 - 1.66T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 - 4.34T + 37T^{2} \) |
| 41 | \( 1 - 6.95T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 0.637T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−7.897469930018212520957706226095, −7.49339666980887750034506821176, −6.39779238876509865027056648751, −5.51370137570114927097568343727, −5.06489012155058622744354273476, −4.36454231660166562744923572010, −3.52700359740767183721304744288, −2.70100880977030274700456007458, −1.53059653701221511146746739057, −0.959429761585262024096309293536,
0.959429761585262024096309293536, 1.53059653701221511146746739057, 2.70100880977030274700456007458, 3.52700359740767183721304744288, 4.36454231660166562744923572010, 5.06489012155058622744354273476, 5.51370137570114927097568343727, 6.39779238876509865027056648751, 7.49339666980887750034506821176, 7.897469930018212520957706226095
