| L(s) = 1 | − i·2-s + 0.732·3-s + 4-s − 3.73i·5-s − 0.732i·6-s − 3.46·7-s − 3i·8-s − 2.46·9-s − 3.73·10-s − 1.26·11-s + 0.732·12-s + 3.46i·13-s + 3.46i·14-s − 2.73i·15-s − 16-s + 0.267i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.422·3-s + 0.5·4-s − 1.66i·5-s − 0.298i·6-s − 1.30·7-s − 1.06i·8-s − 0.821·9-s − 1.18·10-s − 0.382·11-s + 0.211·12-s + 0.960i·13-s + 0.925i·14-s − 0.705i·15-s − 0.250·16-s + 0.0649i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008071963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008071963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 3.73iT - 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 0.267iT - 17T^{2} \) |
| 19 | \( 1 + 4.73iT - 19T^{2} \) |
| 23 | \( 1 - 2.73iT - 23T^{2} \) |
| 29 | \( 1 - 3.73iT - 29T^{2} \) |
| 31 | \( 1 - 2.19iT - 31T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + 6.53iT - 59T^{2} \) |
| 61 | \( 1 + 7.73iT - 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 14.1iT - 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 1.19iT - 89T^{2} \) |
| 97 | \( 1 + 7.73iT - 97T^{2} \) |
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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
−9.255942731813928760969287213497, −8.648919610684700161906919088832, −7.55799874561298395566952545771, −6.61805391903578674833830691395, −5.74827122237514731922574135456, −4.72468665671027544177895348357, −3.65327875715627405366023401269, −2.82537727032225578153737419599, −1.72906199465194543505727267895, −0.34594861081684652589870252294,
2.53991944082573632313593927268, 2.83627263380730820393608067854, 3.73112188720031586997673759384, 5.64008676262438268858817248066, 6.10192627438941872242619254154, 6.72582939769427304722637418285, 7.65949694647793177279208122016, 8.040401627773339851673663518209, 9.253164704194101797576830322070, 10.23616360822771018764884669096
