| L(s) = 1 | + (−2.56 + 2.56i)3-s + (99.7 + 99.7i)7-s + 229. i·9-s − 688. i·11-s + (−296. − 296. i)13-s + (−204. + 204. i)17-s − 1.28e3·19-s − 511.·21-s + (−1.87e3 + 1.87e3i)23-s + (−1.21e3 − 1.21e3i)27-s − 2.39e3i·29-s + 4.82e3i·31-s + (1.76e3 + 1.76e3i)33-s + (5.89e3 − 5.89e3i)37-s + 1.52e3·39-s + ⋯ |
| L(s) = 1 | + (−0.164 + 0.164i)3-s + (0.769 + 0.769i)7-s + 0.945i·9-s − 1.71i·11-s + (−0.486 − 0.486i)13-s + (−0.171 + 0.171i)17-s − 0.816·19-s − 0.253·21-s + (−0.737 + 0.737i)23-s + (−0.320 − 0.320i)27-s − 0.528i·29-s + 0.901i·31-s + (0.282 + 0.282i)33-s + (0.707 − 0.707i)37-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7219799016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7219799016\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (2.56 - 2.56i)T - 243iT^{2} \) |
| 7 | \( 1 + (-99.7 - 99.7i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 688. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (296. + 296. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (204. - 204. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 1.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.87e3 - 1.87e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 2.39e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.82e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-5.89e3 + 5.89e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 2.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.17e4 + 1.17e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.57e4 - 1.57e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.68e4 + 1.68e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 651.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.45e4 + 3.45e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 503. iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (2.09e4 + 2.09e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 5.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (2.25e4 - 2.25e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 4.99e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.18e5 + 1.18e5i)T - 8.58e9iT^{2} \) |
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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
−10.43141003297651424030555099067, −9.083014124824996057269721338726, −8.307484954650117045428514905699, −7.63038346972598834335597627280, −6.01321346978445962241676956871, −5.46645489906710584290603572128, −4.34938029247780360673819032104, −2.92344093578523654318750255691, −1.80819965249135439589128742253, −0.18009509660166193350006167671,
1.25179122988758692343421476934, 2.37271976116337211911862953132, 4.18118142244053849073168843184, 4.61212435931759217015077206215, 6.17274198413262666121150143645, 7.08218402751407259964588607646, 7.76323150295778537604954603017, 9.035433121547815555026538055213, 9.876662371654609520216397738818, 10.70132092898974383913896129990
