L(s) = 1 | + (32 − 32i)2-s + (693. + 693. i)3-s − 2.04e3i·4-s + (7.74e3 + 1.35e4i)5-s + 4.43e4·6-s + (−1.08e5 + 1.08e5i)7-s + (−6.55e4 − 6.55e4i)8-s + 4.29e5i·9-s + (6.82e5 + 1.86e5i)10-s + 2.46e6·11-s + (1.41e6 − 1.41e6i)12-s + (7.82e5 + 7.82e5i)13-s + 6.94e6i·14-s + (−4.03e6 + 1.47e7i)15-s − 4.19e6·16-s + (2.16e7 − 2.16e7i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.950 + 0.950i)3-s − 0.5i·4-s + (0.495 + 0.868i)5-s + 0.950·6-s + (−0.922 + 0.922i)7-s + (−0.250 − 0.250i)8-s + 0.808i·9-s + (0.682 + 0.186i)10-s + 1.39·11-s + (0.475 − 0.475i)12-s + (0.162 + 0.162i)13-s + 0.922i·14-s + (−0.354 + 1.29i)15-s − 0.250·16-s + (0.895 − 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(2.77699 + 1.09389i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.77699 + 1.09389i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-32 + 32i)T \) |
| 5 | \( 1 + (-7.74e3 - 1.35e4i)T \) |
good | 3 | \( 1 + (-693. - 693. i)T + 5.31e5iT^{2} \) |
| 7 | \( 1 + (1.08e5 - 1.08e5i)T - 1.38e10iT^{2} \) |
| 11 | \( 1 - 2.46e6T + 3.13e12T^{2} \) |
| 13 | \( 1 + (-7.82e5 - 7.82e5i)T + 2.32e13iT^{2} \) |
| 17 | \( 1 + (-2.16e7 + 2.16e7i)T - 5.82e14iT^{2} \) |
| 19 | \( 1 - 1.52e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + (2.05e8 + 2.05e8i)T + 2.19e16iT^{2} \) |
| 29 | \( 1 + 2.06e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 + 1.07e8T + 7.87e17T^{2} \) |
| 37 | \( 1 + (-1.52e8 + 1.52e8i)T - 6.58e18iT^{2} \) |
| 41 | \( 1 - 7.91e9T + 2.25e19T^{2} \) |
| 43 | \( 1 + (5.63e9 + 5.63e9i)T + 3.99e19iT^{2} \) |
| 47 | \( 1 + (-2.56e9 + 2.56e9i)T - 1.16e20iT^{2} \) |
| 53 | \( 1 + (1.46e10 + 1.46e10i)T + 4.91e20iT^{2} \) |
| 59 | \( 1 - 5.45e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 + 1.89e9T + 2.65e21T^{2} \) |
| 67 | \( 1 + (5.73e10 - 5.73e10i)T - 8.18e21iT^{2} \) |
| 71 | \( 1 - 1.06e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (5.40e10 + 5.40e10i)T + 2.29e22iT^{2} \) |
| 79 | \( 1 + 7.36e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (-3.76e11 - 3.76e11i)T + 1.06e23iT^{2} \) |
| 89 | \( 1 + 4.32e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + (-9.08e11 + 9.08e11i)T - 6.93e23iT^{2} \) |
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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
−18.51098138354754953270731656272, −16.15853099005251925033899001006, −14.77557639524702213691685940648, −14.04426344465367613119279231391, −12.03863068174093331963221000087, −10.07082656710922486120536858584, −9.177446929154321283771888924477, −6.20105399245093140862734670127, −3.75844889258990611288352339447, −2.54224139153016316943385007744,
1.36819003458998788403911906946, 3.70298978436624568070546671769, 6.29043934589467283317071053813, 7.83950051754511779914798100506, 9.430332261412995799504297113276, 12.43634211877720896198455027624, 13.43957609558400373527157010389, 14.30604006430513612356641993041, 16.33293176696235782132801983947, 17.46585303315520556329340680276