| L(s) = 1 | − 5-s − 0.113·7-s − 3.39·11-s + 6.27·13-s − 3.61·17-s − 1.39·19-s − 23-s + 25-s + 6.38·29-s + 9.45·31-s + 0.113·35-s − 7.78·37-s − 11.0·41-s + 1.55·43-s − 1.70·47-s − 6.98·49-s + 8.71·53-s + 3.39·55-s + 8.95·59-s − 1.83·61-s − 6.27·65-s + 9.21·67-s + 6.82·71-s − 2.27·73-s + 0.387·77-s + 11.7·79-s − 1.16·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.0430·7-s − 1.02·11-s + 1.74·13-s − 0.877·17-s − 0.321·19-s − 0.208·23-s + 0.200·25-s + 1.18·29-s + 1.69·31-s + 0.0192·35-s − 1.28·37-s − 1.72·41-s + 0.237·43-s − 0.248·47-s − 0.998·49-s + 1.19·53-s + 0.458·55-s + 1.16·59-s − 0.234·61-s − 0.778·65-s + 1.12·67-s + 0.810·71-s − 0.266·73-s + 0.0441·77-s + 1.32·79-s − 0.127·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542007752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542007752\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.113T + 7T^{2} \) |
| 11 | \( 1 + 3.39T + 11T^{2} \) |
| 13 | \( 1 - 6.27T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 - 9.45T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 - 8.71T + 53T^{2} \) |
| 59 | \( 1 - 8.95T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 + 2.27T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 1.16T + 83T^{2} \) |
| 89 | \( 1 - 1.77T + 89T^{2} \) |
| 97 | \( 1 + 0.131T + 97T^{2} \) |
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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
−8.453029916113806364068511724790, −7.913937187168915828813180784055, −6.72821316575325459541665485805, −6.44970009053304233000890405342, −5.38010373692381803314529641614, −4.65960319288485828245121282725, −3.79174508532790750144771235069, −3.02971922407385045009529678351, −1.98465952860745940031424126558, −0.70198113341966677257745759571,
0.70198113341966677257745759571, 1.98465952860745940031424126558, 3.02971922407385045009529678351, 3.79174508532790750144771235069, 4.65960319288485828245121282725, 5.38010373692381803314529641614, 6.44970009053304233000890405342, 6.72821316575325459541665485805, 7.913937187168915828813180784055, 8.453029916113806364068511724790
