| L(s) = 1 | + (−0.509 + 1.34i)2-s + (0.568 + 0.822i)3-s + (−0.0481 − 0.0426i)4-s + (−0.0880 + 0.724i)5-s + (−1.39 + 0.343i)6-s + (−4.13 + 2.17i)7-s + (−2.46 + 1.29i)8-s + (−0.354 + 0.935i)9-s + (−0.928 − 0.487i)10-s + (−0.821 − 2.16i)11-s + (0.00776 − 0.0639i)12-s + (−0.931 − 3.48i)13-s + (−0.808 − 6.66i)14-s + (−0.646 + 0.339i)15-s + (−0.497 − 4.09i)16-s + (−1.28 + 0.675i)17-s + ⋯ |
| L(s) = 1 | + (−0.360 + 0.949i)2-s + (0.327 + 0.475i)3-s + (−0.0240 − 0.0213i)4-s + (−0.0393 + 0.324i)5-s + (−0.569 + 0.140i)6-s + (−1.56 + 0.820i)7-s + (−0.870 + 0.456i)8-s + (−0.118 + 0.311i)9-s + (−0.293 − 0.154i)10-s + (−0.247 − 0.653i)11-s + (0.00224 − 0.0184i)12-s + (−0.258 − 0.966i)13-s + (−0.216 − 1.78i)14-s + (−0.166 + 0.0876i)15-s + (−0.124 − 1.02i)16-s + (−0.312 + 0.163i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.264158 - 0.639945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.264158 - 0.639945i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.568 - 0.822i)T \) |
| 13 | \( 1 + (0.931 + 3.48i)T \) |
| good | 2 | \( 1 + (0.509 - 1.34i)T + (-1.49 - 1.32i)T^{2} \) |
| 5 | \( 1 + (0.0880 - 0.724i)T + (-4.85 - 1.19i)T^{2} \) |
| 7 | \( 1 + (4.13 - 2.17i)T + (3.97 - 5.76i)T^{2} \) |
| 11 | \( 1 + (0.821 + 2.16i)T + (-8.23 + 7.29i)T^{2} \) |
| 17 | \( 1 + (1.28 - 0.675i)T + (9.65 - 13.9i)T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + (3.46 - 9.14i)T + (-21.7 - 19.2i)T^{2} \) |
| 31 | \( 1 + (7.12 - 1.75i)T + (27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (1.60 - 0.395i)T + (32.7 - 17.1i)T^{2} \) |
| 41 | \( 1 + (-2.02 - 2.93i)T + (-14.5 + 38.3i)T^{2} \) |
| 43 | \( 1 + (-9.35 - 2.30i)T + (38.0 + 19.9i)T^{2} \) |
| 47 | \( 1 + (9.30 - 8.24i)T + (5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (-0.773 + 0.406i)T + (30.1 - 43.6i)T^{2} \) |
| 59 | \( 1 + (0.500 - 4.12i)T + (-57.2 - 14.1i)T^{2} \) |
| 61 | \( 1 + (0.434 + 0.227i)T + (34.6 + 50.2i)T^{2} \) |
| 67 | \( 1 + (9.66 - 8.56i)T + (8.07 - 66.5i)T^{2} \) |
| 71 | \( 1 + (5.89 + 8.53i)T + (-25.1 + 66.3i)T^{2} \) |
| 73 | \( 1 + (-0.355 - 0.938i)T + (-54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (3.46 - 3.06i)T + (9.52 - 78.4i)T^{2} \) |
| 83 | \( 1 + (-1.05 + 1.52i)T + (-29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 + 9.35T + 89T^{2} \) |
| 97 | \( 1 + (-0.765 - 6.30i)T + (-94.1 + 23.2i)T^{2} \) |
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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
−11.33813440480419892961584121042, −10.36248628004199233842498531917, −9.314466540828776938962122468812, −8.929908489757688160673211201635, −7.77856734917510790414256547123, −7.01416485030831618447810829609, −5.93969884866411137385117339212, −5.37707108511901139758658346852, −3.19952646374194021272985041528, −3.00442081703399977563877896413,
0.41412504327705214620673083200, 1.97170965154392320711300542354, 3.13646066197273546564570729384, 4.12026343271216046574375531224, 5.84801397900810234944558028229, 6.88638454878803985650309897917, 7.46895228067895564251480495375, 9.058218061315883041315735988426, 9.601279726530433932680644260754, 10.14525254961248383991379451792
