| L(s) = 1 | − 0.979i·2-s + 20.7i·3-s + 31.0·4-s + 20.3·6-s + 233. i·7-s − 61.7i·8-s − 188.·9-s + 621.·11-s + 645. i·12-s + 169i·13-s + 229.·14-s + 932.·16-s − 287. i·17-s + 185. i·18-s + 2.63e3·19-s + ⋯ |
| L(s) = 1 | − 0.173i·2-s + 1.33i·3-s + 0.970·4-s + 0.230·6-s + 1.80i·7-s − 0.341i·8-s − 0.777·9-s + 1.54·11-s + 1.29i·12-s + 0.277i·13-s + 0.312·14-s + 0.910·16-s − 0.241i·17-s + 0.134i·18-s + 1.67·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.296709505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.296709505\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 + 0.979iT - 32T^{2} \) |
| 3 | \( 1 - 20.7iT - 243T^{2} \) |
| 7 | \( 1 - 233. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 621.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 287. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.63e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.16e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.72e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 6.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.66e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.96e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.17e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.21e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.05e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.89e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.63e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.39e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.17e4iT - 8.58e9T^{2} \) |
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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
−11.42933116723221701689425382394, −9.907777083612065547924707188597, −9.452113743394712015930586541737, −8.616230365670838013039490768157, −7.12768673117038360233446024931, −5.97502435262321941565315612826, −5.20749665453848066017032494680, −3.78133814753018525322969028379, −2.86838968975088413434905600381, −1.55174508312849642334074035703,
0.995725041551171598062425769368, 1.33855443233139415546544713138, 2.98492849191074524342747583336, 4.26386671876544906782270772070, 6.09369214696144003248485699046, 6.77658661788774618546610641965, 7.40166582199878225720261890233, 8.041822632914343321998881232948, 9.635434092614960902069930074413, 10.66291933283943845715751527035
