| L(s) = 1 | + 6.26·2-s − 23.5i·3-s − 24.6·4-s − 124.·5-s − 147. i·6-s − 222.·7-s − 556.·8-s + 176.·9-s − 780.·10-s + 1.14e3i·11-s + 580. i·12-s − 3.58e3i·13-s − 1.39e3·14-s + 2.92e3i·15-s − 1.90e3·16-s − 7.39e3i·17-s + ⋯ |
| L(s) = 1 | + 0.783·2-s − 0.870i·3-s − 0.385·4-s − 0.996·5-s − 0.682i·6-s − 0.649·7-s − 1.08·8-s + 0.242·9-s − 0.780·10-s + 0.862i·11-s + 0.335i·12-s − 1.63i·13-s − 0.509·14-s + 0.867i·15-s − 0.465·16-s − 1.50i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0925401 - 0.771578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0925401 - 0.771578i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-2.89e4 + 7.04e3i)T \) |
| good | 2 | \( 1 - 6.26T + 64T^{2} \) |
| 3 | \( 1 + 23.5iT - 729T^{2} \) |
| 5 | \( 1 + 124.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 222.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.14e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.58e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 7.39e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.49e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.03e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 3.39e4iT - 5.94e8T^{2} \) |
| 37 | \( 1 - 4.46e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 6.74e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 3.63e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.04e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.03e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 4.03e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.05e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 1.51e3T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.47e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 3.64e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 1.77e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 6.76e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 8.43e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.64e6T + 8.32e11T^{2} \) |
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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
−15.01520199727406078075844931596, −13.48835582847344963258361814242, −12.69488391775775176731597940319, −11.91419111930557767485089518537, −9.873693904535791604923574574392, −8.069343656258649277860023012142, −6.79799501066190825740864305828, −4.91177113810894459952927055383, −3.22179798876921965032244420484, −0.32830159169718150813392847365,
3.67682913233144744837109168929, 4.37883289938726039417469355165, 6.30410385241192917603383870729, 8.483819816338030662072830762875, 9.751519042340679288947961204975, 11.33077803038252413716219779486, 12.55304201204417248757132847011, 13.80425717064837378199985229312, 15.04647658453229061776506313022, 15.86860567314338304372364366173
