| L(s) = 1 | + (64 − 64i)2-s + (1.76e3 + 1.76e3i)3-s − 8.19e3i·4-s + 2.25e5·6-s + (5.96e4 − 5.96e4i)7-s + (−5.24e5 − 5.24e5i)8-s + 1.41e6i·9-s − 4.64e6·11-s + (1.44e7 − 1.44e7i)12-s + (4.85e7 + 4.85e7i)13-s − 7.63e6i·14-s − 6.71e7·16-s + (4.70e8 − 4.70e8i)17-s + (9.06e7 + 9.06e7i)18-s + 7.64e8i·19-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.5i)2-s + (0.805 + 0.805i)3-s − 0.5i·4-s + 0.805·6-s + (0.0724 − 0.0724i)7-s + (−0.250 − 0.250i)8-s + 0.296i·9-s − 0.238·11-s + (0.402 − 0.402i)12-s + (0.773 + 0.773i)13-s − 0.0724i·14-s − 0.250·16-s + (1.14 − 1.14i)17-s + (0.148 + 0.148i)18-s + 0.855i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(4.131436733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.131436733\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-64 + 64i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1.76e3 - 1.76e3i)T + 4.78e6iT^{2} \) |
| 7 | \( 1 + (-5.96e4 + 5.96e4i)T - 6.78e11iT^{2} \) |
| 11 | \( 1 + 4.64e6T + 3.79e14T^{2} \) |
| 13 | \( 1 + (-4.85e7 - 4.85e7i)T + 3.93e15iT^{2} \) |
| 17 | \( 1 + (-4.70e8 + 4.70e8i)T - 1.68e17iT^{2} \) |
| 19 | \( 1 - 7.64e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + (1.43e9 + 1.43e9i)T + 1.15e19iT^{2} \) |
| 29 | \( 1 - 3.55e9iT - 2.97e20T^{2} \) |
| 31 | \( 1 - 4.07e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + (-1.25e11 + 1.25e11i)T - 9.01e21iT^{2} \) |
| 41 | \( 1 + 1.93e11T + 3.79e22T^{2} \) |
| 43 | \( 1 + (-2.99e11 - 2.99e11i)T + 7.38e22iT^{2} \) |
| 47 | \( 1 + (4.49e11 - 4.49e11i)T - 2.56e23iT^{2} \) |
| 53 | \( 1 + (-8.73e11 - 8.73e11i)T + 1.37e24iT^{2} \) |
| 59 | \( 1 + 2.30e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 1.81e12T + 9.87e24T^{2} \) |
| 67 | \( 1 + (-5.96e12 + 5.96e12i)T - 3.67e25iT^{2} \) |
| 71 | \( 1 - 5.96e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + (-7.19e11 - 7.19e11i)T + 1.22e26iT^{2} \) |
| 79 | \( 1 + 3.29e12iT - 3.68e26T^{2} \) |
| 83 | \( 1 + (-6.80e12 - 6.80e12i)T + 7.36e26iT^{2} \) |
| 89 | \( 1 + 3.40e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-7.87e13 + 7.87e13i)T - 6.52e27iT^{2} \) |
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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
−12.50771756989980483214151314177, −11.36326150731701405853411363721, −10.08236671032131441647365047234, −9.288490110073637943613295642636, −7.961133694556783015279127266978, −6.16466133081584125036110438563, −4.64266363607133122206165116795, −3.64321614431736290604003123611, −2.59702787713815172326351091392, −1.00714613095134617534798575820,
1.05606968621561111064847637759, 2.49208968901108364352731741620, 3.64585600581746410677727956004, 5.34327593700337202662661444670, 6.60584855008472342265836562586, 7.924055098345665799966260783015, 8.443636199225214514427507715739, 10.23038914301854065423844875077, 11.81764976784832298812639819358, 13.06728617416063034615567827365
