| L(s) = 1 | − 5-s − 2·7-s − 4.44·11-s − 13-s − 6.89·17-s − 0.449·19-s + 6.44·23-s + 25-s − 4·29-s + 4.44·31-s + 2·35-s + 4.89·37-s − 10.8·41-s − 11.3·43-s − 2·47-s − 3·49-s − 1.10·53-s + 4.44·55-s + 9.34·59-s − 5.79·61-s + 65-s + 5.10·67-s − 3.55·71-s − 14.6·73-s + 8.89·77-s − 4.89·79-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.34·11-s − 0.277·13-s − 1.67·17-s − 0.103·19-s + 1.34·23-s + 0.200·25-s − 0.742·29-s + 0.799·31-s + 0.338·35-s + 0.805·37-s − 1.70·41-s − 1.73·43-s − 0.291·47-s − 0.428·49-s − 0.151·53-s + 0.599·55-s + 1.21·59-s − 0.742·61-s + 0.124·65-s + 0.623·67-s − 0.421·71-s − 1.72·73-s + 1.01·77-s − 0.551·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5788289019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5788289019\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 + 0.449T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 + 5.79T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−7.70727237184113514608136864128, −6.85125252822541008203862417707, −6.63310963620892370828605445276, −5.56803387310248850463965791361, −4.89636875718957182353661540301, −4.32275942810037946211135355664, −3.24208821229344341166939280816, −2.79735866783702788274398438805, −1.83061995344127152210578849019, −0.34377221654586199444786959032,
0.34377221654586199444786959032, 1.83061995344127152210578849019, 2.79735866783702788274398438805, 3.24208821229344341166939280816, 4.32275942810037946211135355664, 4.89636875718957182353661540301, 5.56803387310248850463965791361, 6.63310963620892370828605445276, 6.85125252822541008203862417707, 7.70727237184113514608136864128
