| L(s) = 1 | + 3.00e3i·3-s − 3.49e4i·5-s + (−2.09e5 + 7.96e5i)7-s − 4.27e6·9-s − 3.27e7·11-s − 2.83e6i·13-s + 1.05e8·15-s − 1.16e8i·17-s + 7.25e8i·19-s + (−2.39e9 − 6.30e8i)21-s − 5.44e9·23-s − 1.22e9·25-s + 1.54e9i·27-s − 2.49e10·29-s + 1.90e10i·31-s + ⋯ |
| L(s) = 1 | + 1.37i·3-s − 0.447i·5-s + (−0.254 + 0.967i)7-s − 0.892·9-s − 1.67·11-s − 0.0452i·13-s + 0.615·15-s − 0.284i·17-s + 0.811i·19-s + (−1.33 − 0.349i)21-s − 1.59·23-s − 0.199·25-s + 0.147i·27-s − 1.44·29-s + 0.693i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.3759389358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3759389358\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.49e4iT \) |
| 7 | \( 1 + (2.09e5 - 7.96e5i)T \) |
| good | 3 | \( 1 - 3.00e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 + 3.27e7T + 3.79e14T^{2} \) |
| 13 | \( 1 + 2.83e6iT - 3.93e15T^{2} \) |
| 17 | \( 1 + 1.16e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 7.25e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 5.44e9T + 1.15e19T^{2} \) |
| 29 | \( 1 + 2.49e10T + 2.97e20T^{2} \) |
| 31 | \( 1 - 1.90e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 - 1.63e11T + 9.01e21T^{2} \) |
| 41 | \( 1 - 6.25e9iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 1.86e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 1.72e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 1.06e12T + 1.37e24T^{2} \) |
| 59 | \( 1 + 8.95e10iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 1.59e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 4.05e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 1.01e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + 2.29e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 + 2.61e13T + 3.68e26T^{2} \) |
| 83 | \( 1 + 3.43e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 3.86e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.82e13iT - 6.52e27T^{2} \) |
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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
−10.24646220822355838673827831958, −9.703363599688231644379037959200, −8.657313195797545918202551355132, −7.74346273007750653645825662975, −5.83427933098695901194462712069, −5.27001434055292893271690906939, −4.21416821398210627951848786216, −3.10135730148950822411724169199, −2.01506350626494266312210004006, −0.10370176302097290881245980531,
0.61002964011141836707329654558, 1.91241685215468234075320509787, 2.74473666570682074276547687097, 4.12797068459429860332684761783, 5.66454417569193612411041993617, 6.64708534894697393865404792842, 7.60861065641318784070382835598, 7.988365346116804902662269921552, 9.744695720120434037456963589839, 10.68553087511279574591131108013
