L(s) = 1 | + (4.66 − 2.28i)3-s + (11.5 − 11.5i)5-s + 0.829·7-s + (16.5 − 21.3i)9-s + (38.2 + 38.2i)11-s + (20.0 − 20.0i)13-s + (27.4 − 80.0i)15-s + 77.9i·17-s + (37.6 + 37.6i)19-s + (3.87 − 1.89i)21-s − 159. i·23-s − 140. i·25-s + (28.6 − 137. i)27-s + (−26.6 − 26.6i)29-s + 226. i·31-s + ⋯ |
L(s) = 1 | + (0.898 − 0.439i)3-s + (1.02 − 1.02i)5-s + 0.0447·7-s + (0.613 − 0.789i)9-s + (1.04 + 1.04i)11-s + (0.427 − 0.427i)13-s + (0.472 − 1.37i)15-s + 1.11i·17-s + (0.455 + 0.455i)19-s + (0.0402 − 0.0196i)21-s − 1.44i·23-s − 1.12i·25-s + (0.204 − 0.978i)27-s + (−0.170 − 0.170i)29-s + 1.31i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.441942101\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.441942101\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.66 + 2.28i)T \) |
good | 5 | \( 1 + (-11.5 + 11.5i)T - 125iT^{2} \) |
| 7 | \( 1 - 0.829T + 343T^{2} \) |
| 11 | \( 1 + (-38.2 - 38.2i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-20.0 + 20.0i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 77.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-37.6 - 37.6i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 159. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (26.6 + 26.6i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 226. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (176. + 176. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 301.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-96.1 + 96.1i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (138. - 138. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-168. - 168. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-178. + 178. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-233. - 233. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 668. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.21e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 39.4iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-293. + 293. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 451.T + 9.12e5T^{2} \) |
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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
−10.44825347604710902379807538718, −9.686218207269449967747432614445, −8.850986986028995655698244561573, −8.289246341227089428194774522214, −6.96370039163724353581498712976, −6.08245528859756953631635750649, −4.79900059115077161303777055050, −3.64450287776278107316705151545, −1.98924509596010957574038386821, −1.26146160309985536090265413853,
1.60267390998114960894463568328, 2.89695995865521484850422338918, 3.69217913528413819404716666760, 5.23776576104535444095897532245, 6.38811396780407332717000454449, 7.22466143705937158591557054620, 8.464076655844917867667669666941, 9.485682789061896499074436854346, 9.783714345074460238995708794242, 11.09144013624748434944122658576