| L(s) = 1 | − 0.870·3-s − 1.04·5-s − 2.24·9-s − 0.892·11-s − 2.84·13-s + 0.906·15-s + 1.88·17-s + 19-s + 0.673·23-s − 3.91·25-s + 4.56·27-s − 7.93·29-s − 7.13·31-s + 0.776·33-s + 11.1·37-s + 2.47·39-s + 2.64·41-s − 11.8·43-s + 2.33·45-s − 6.95·47-s − 1.63·51-s − 2.16·53-s + 0.929·55-s − 0.870·57-s + 2.98·59-s + 13.1·61-s + 2.96·65-s + ⋯ |
| L(s) = 1 | − 0.502·3-s − 0.465·5-s − 0.747·9-s − 0.269·11-s − 0.789·13-s + 0.234·15-s + 0.456·17-s + 0.229·19-s + 0.140·23-s − 0.782·25-s + 0.878·27-s − 1.47·29-s − 1.28·31-s + 0.135·33-s + 1.83·37-s + 0.396·39-s + 0.412·41-s − 1.80·43-s + 0.348·45-s − 1.01·47-s − 0.229·51-s − 0.297·53-s + 0.125·55-s − 0.115·57-s + 0.388·59-s + 1.68·61-s + 0.367·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6587396520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6587396520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.870T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 + 0.892T + 11T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 23 | \( 1 - 0.673T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 7.13T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 6.95T + 47T^{2} \) |
| 53 | \( 1 + 2.16T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 + 0.0366T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 - 2.88T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 - 0.938T + 97T^{2} \) |
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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
−7.88575440319804913387746542611, −7.25513115439948945486609460533, −6.50414137021468795329680815565, −5.56562481826564744064463852889, −5.33545949082283432808251791029, −4.35575187950999687636190875283, −3.54638231906548868516789202596, −2.76538722790967016194683311890, −1.78026030357959548222577188482, −0.40349134605367861473004848876,
0.40349134605367861473004848876, 1.78026030357959548222577188482, 2.76538722790967016194683311890, 3.54638231906548868516789202596, 4.35575187950999687636190875283, 5.33545949082283432808251791029, 5.56562481826564744064463852889, 6.50414137021468795329680815565, 7.25513115439948945486609460533, 7.88575440319804913387746542611
