| L(s) = 1 | + 2.55·3-s − 7.40·5-s − 10.4·7-s − 20.4·9-s − 48.7·11-s − 4.45·13-s − 18.8·15-s − 61.5·17-s − 19·19-s − 26.7·21-s + 34.8·23-s − 70.2·25-s − 121.·27-s + 179.·29-s + 132.·31-s − 124.·33-s + 77.6·35-s + 99.3·37-s − 11.3·39-s + 382.·41-s + 129.·43-s + 151.·45-s − 488.·47-s − 232.·49-s − 157.·51-s + 129.·53-s + 361.·55-s + ⋯ |
| L(s) = 1 | + 0.491·3-s − 0.661·5-s − 0.566·7-s − 0.758·9-s − 1.33·11-s − 0.0950·13-s − 0.325·15-s − 0.877·17-s − 0.229·19-s − 0.278·21-s + 0.315·23-s − 0.561·25-s − 0.864·27-s + 1.14·29-s + 0.769·31-s − 0.657·33-s + 0.375·35-s + 0.441·37-s − 0.0466·39-s + 1.45·41-s + 0.459·43-s + 0.502·45-s − 1.51·47-s − 0.678·49-s − 0.431·51-s + 0.334·53-s + 0.885·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9918229364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9918229364\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 2.55T + 27T^{2} \) |
| 5 | \( 1 + 7.40T + 125T^{2} \) |
| 7 | \( 1 + 10.4T + 343T^{2} \) |
| 11 | \( 1 + 48.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.45T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 34.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 382.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 129.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 479.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 394.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 19.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 221.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 258.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 469.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 373.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.65e3T + 9.12e5T^{2} \) |
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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
−9.320285526754862018381747413562, −8.290305318245967101542994502984, −8.028962115283938990495439858520, −6.95133631117507028442390699027, −6.05408609261830769685586573277, −5.04189107867896207970331359986, −4.06369228919620192884858262436, −2.98205980794848859406404583219, −2.37975345206888418625792237048, −0.46423832972324917141660534403,
0.46423832972324917141660534403, 2.37975345206888418625792237048, 2.98205980794848859406404583219, 4.06369228919620192884858262436, 5.04189107867896207970331359986, 6.05408609261830769685586573277, 6.95133631117507028442390699027, 8.028962115283938990495439858520, 8.290305318245967101542994502984, 9.320285526754862018381747413562
