| L(s) = 1 | + (−429. + 429. i)3-s + (2.47e3 − 2.47e3i)5-s + 1.98e5·7-s + 1.62e5i·9-s + (−1.13e6 − 1.13e6i)11-s + (−1.22e6 − 1.22e6i)13-s + 2.12e6i·15-s − 8.93e5·17-s + (2.55e7 − 2.55e7i)19-s + (−8.53e7 + 8.53e7i)21-s − 2.41e8·23-s + 2.31e8i·25-s + (−2.98e8 − 2.98e8i)27-s + (−1.87e8 − 1.87e8i)29-s + 9.02e8i·31-s + ⋯ |
| L(s) = 1 | + (−0.589 + 0.589i)3-s + (0.158 − 0.158i)5-s + 1.68·7-s + 0.305i·9-s + (−0.638 − 0.638i)11-s + (−0.253 − 0.253i)13-s + 0.186i·15-s − 0.0370·17-s + (0.542 − 0.542i)19-s + (−0.995 + 0.995i)21-s − 1.63·23-s + 0.949i·25-s + (−0.769 − 0.769i)27-s + (−0.315 − 0.315i)29-s + 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.7031033902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7031033902\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (429. - 429. i)T - 5.31e5iT^{2} \) |
| 5 | \( 1 + (-2.47e3 + 2.47e3i)T - 2.44e8iT^{2} \) |
| 7 | \( 1 - 1.98e5T + 1.38e10T^{2} \) |
| 11 | \( 1 + (1.13e6 + 1.13e6i)T + 3.13e12iT^{2} \) |
| 13 | \( 1 + (1.22e6 + 1.22e6i)T + 2.32e13iT^{2} \) |
| 17 | \( 1 + 8.93e5T + 5.82e14T^{2} \) |
| 19 | \( 1 + (-2.55e7 + 2.55e7i)T - 2.21e15iT^{2} \) |
| 23 | \( 1 + 2.41e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (1.87e8 + 1.87e8i)T + 3.53e17iT^{2} \) |
| 31 | \( 1 - 9.02e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 + (-1.02e9 + 1.02e9i)T - 6.58e18iT^{2} \) |
| 41 | \( 1 + 6.51e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + (-6.20e9 - 6.20e9i)T + 3.99e19iT^{2} \) |
| 47 | \( 1 - 1.35e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + (9.75e9 - 9.75e9i)T - 4.91e20iT^{2} \) |
| 59 | \( 1 + (2.24e10 + 2.24e10i)T + 1.77e21iT^{2} \) |
| 61 | \( 1 + (4.44e9 + 4.44e9i)T + 2.65e21iT^{2} \) |
| 67 | \( 1 + (5.79e10 - 5.79e10i)T - 8.18e21iT^{2} \) |
| 71 | \( 1 - 2.01e11T + 1.64e22T^{2} \) |
| 73 | \( 1 - 1.84e11iT - 2.29e22T^{2} \) |
| 79 | \( 1 - 2.43e11iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (6.52e10 - 6.52e10i)T - 1.06e23iT^{2} \) |
| 89 | \( 1 + 2.19e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 4.30e11T + 6.93e23T^{2} \) |
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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
−11.13939634207237699182828366039, −10.86026385828337925533122597839, −9.593053644954369838011064592277, −8.254845500507319495603134291836, −7.57289632967029445824514402157, −5.69851587590204083270769326057, −5.13432631776947538895689554415, −4.16013891064751597230776707991, −2.44361832229261627280738673234, −1.24915595049584972212297976657,
0.15662564673413265705122525523, 1.44559174917988336229278258212, 2.24551299825591871048490927301, 4.13071021421796453194106513535, 5.20839505137246505141843022250, 6.20948179623860387138244835479, 7.49501509477591206797385045781, 8.137461389525780896522268481472, 9.651656095542123405104672914680, 10.73452109568085809070694420944
