| L(s) = 1 | + 3·3-s + 6.30·5-s + 9·9-s − 48.9·11-s + 2.60·13-s + 18.9·15-s − 136.·17-s + 45.2·19-s + 38.1·23-s − 85.2·25-s + 27·27-s + 52.7·29-s − 14.7·31-s − 146.·33-s + 333.·37-s + 7.82·39-s − 227.·41-s + 398.·43-s + 56.7·45-s − 184.·47-s − 410.·51-s + 359.·53-s − 308.·55-s + 135.·57-s + 99.9·59-s + 674.·61-s + 16.4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.563·5-s + 0.333·9-s − 1.34·11-s + 0.0556·13-s + 0.325·15-s − 1.95·17-s + 0.545·19-s + 0.345·23-s − 0.682·25-s + 0.192·27-s + 0.337·29-s − 0.0856·31-s − 0.774·33-s + 1.48·37-s + 0.0321·39-s − 0.865·41-s + 1.41·43-s + 0.187·45-s − 0.572·47-s − 1.12·51-s + 0.932·53-s − 0.755·55-s + 0.315·57-s + 0.220·59-s + 1.41·61-s + 0.0313·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.561143045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.561143045\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6.30T + 125T^{2} \) |
| 11 | \( 1 + 48.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.60T + 2.19e3T^{2} \) |
| 17 | \( 1 + 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 184.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 359.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 99.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.18e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 836.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 201.T + 9.12e5T^{2} \) |
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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
−8.575538226071418621300594096551, −7.987728412473259123450125143397, −7.13301480205609200915360921693, −6.36756352232415575144649635388, −5.42609436563531896250073384390, −4.68700379603590535584730063871, −3.71445667165639460204541437344, −2.52315834012646762498768932446, −2.15858186618317234552972511798, −0.66564183129699699194764421106,
0.66564183129699699194764421106, 2.15858186618317234552972511798, 2.52315834012646762498768932446, 3.71445667165639460204541437344, 4.68700379603590535584730063871, 5.42609436563531896250073384390, 6.36756352232415575144649635388, 7.13301480205609200915360921693, 7.987728412473259123450125143397, 8.575538226071418621300594096551
