L(s) = 1 | + (−2.96 − 2.96i)2-s + (−2.12 − 2.12i)3-s + 9.58i·4-s + (−8.64 − 7.08i)5-s + 12.5i·6-s + (−4.10 − 18.0i)7-s + (4.70 − 4.70i)8-s + 8.99i·9-s + (4.63 + 46.6i)10-s − 55.3·11-s + (20.3 − 20.3i)12-s + (43.2 + 43.2i)13-s + (−41.3 + 65.7i)14-s + (3.31 + 33.3i)15-s + 48.7·16-s + (75.6 − 75.6i)17-s + ⋯ |
L(s) = 1 | + (−1.04 − 1.04i)2-s + (−0.408 − 0.408i)3-s + 1.19i·4-s + (−0.773 − 0.633i)5-s + 0.856i·6-s + (−0.221 − 0.975i)7-s + (0.208 − 0.208i)8-s + 0.333i·9-s + (0.146 + 1.47i)10-s − 1.51·11-s + (0.489 − 0.489i)12-s + (0.922 + 0.922i)13-s + (−0.789 + 1.25i)14-s + (0.0570 + 0.574i)15-s + 0.762·16-s + (1.07 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00785597 + 0.00367973i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00785597 + 0.00367973i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.12 + 2.12i)T \) |
| 5 | \( 1 + (8.64 + 7.08i)T \) |
| 7 | \( 1 + (4.10 + 18.0i)T \) |
good | 2 | \( 1 + (2.96 + 2.96i)T + 8iT^{2} \) |
| 11 | \( 1 + 55.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-43.2 - 43.2i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-75.6 + 75.6i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 23.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-25.2 + 25.2i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 280. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 120. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (267. + 267. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 185. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (133. - 133. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (280. - 280. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (99.3 - 99.3i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 357.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 228. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + (-115. - 115. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 71.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-114. - 114. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 58.9iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (145. + 145. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (424. - 424. i)T - 9.12e5iT^{2} \) |
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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
−12.93257701918157730832128494020, −12.19438145206010777033647370683, −11.06815810815800769355985193129, −10.53435611410931922485881987965, −9.192567489471033530614078448541, −8.104737109491615325072499592949, −7.16604569747673440639066394774, −5.11036920319504886504118345750, −3.33350288289232219079674096604, −1.27207372578708959896083218150,
0.007971329775133639210141499629, 3.28739350078312768553970058729, 5.49929216071487133211827519978, 6.35680917596062292790458433662, 7.939028440568531627447520936493, 8.311146234083615250903230500778, 9.925908245877627134247097389818, 10.60386468768830251710486187483, 11.91201278995310877852777983592, 13.09069595004789960779904546708