| L(s) = 1 | + (3.17e3 − 5.49e3i)2-s + (2.39e6 + 4.14e6i)3-s + (2.48e8 + 4.30e8i)4-s + (−5.04e9 + 8.73e9i)5-s + (3.03e10 − 1.90e−6i)6-s + (7.64e11 − 1.62e12i)7-s + (6.55e12 − 0.000488i)8-s + (−1.14e13 + 1.98e13i)9-s + (3.19e13 + 5.54e13i)10-s + (9.18e14 + 1.59e15i)11-s + (−1.18e15 + 2.05e15i)12-s + 2.10e16·13-s + (−6.49e15 − 9.34e15i)14-s − 4.82e16·15-s + (−1.12e17 + 1.94e17i)16-s + (−2.90e17 − 5.02e17i)17-s + ⋯ |
| L(s) = 1 | + (0.136 − 0.237i)2-s + (0.288 + 0.499i)3-s + (0.462 + 0.801i)4-s + (−0.369 + 0.640i)5-s + 0.158·6-s + (0.426 − 0.904i)7-s + 0.526·8-s + (−0.166 + 0.288i)9-s + (0.101 + 0.175i)10-s + (0.729 + 1.26i)11-s + (−0.267 + 0.462i)12-s + 1.47·13-s + (−0.156 − 0.224i)14-s − 0.426·15-s + (−0.390 + 0.676i)16-s + (−0.417 − 0.723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(3.602748276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.602748276\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.39e6 - 4.14e6i)T \) |
| 7 | \( 1 + (-7.64e11 + 1.62e12i)T \) |
| good | 2 | \( 1 + (-3.17e3 + 5.49e3i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (5.04e9 - 8.73e9i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-9.18e14 - 1.59e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 2.10e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (2.90e17 + 5.02e17i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.44e18 - 4.23e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-2.19e19 + 3.81e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 2.92e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (2.38e20 + 4.13e20i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-2.70e22 + 4.69e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 + 2.95e22T + 5.89e46T^{2} \) |
| 43 | \( 1 - 6.49e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-7.71e23 + 1.33e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-4.13e24 - 7.16e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.68e25 - 4.65e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (1.08e24 - 1.87e24i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (2.70e26 + 4.68e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 + 5.57e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-3.77e26 - 6.53e26i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (1.93e27 - 3.34e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 + 2.29e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (5.26e27 - 9.12e27i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 + 7.26e28T + 4.13e57T^{2} \) |
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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
−12.33758915676181782617765217364, −11.13350438153829978819984144681, −10.40205961939567743409499888406, −8.689841340407600498818490005580, −7.50743000473240825804586763106, −6.59643790643885212191005434340, −4.33343726359094780449859201716, −3.83997026800636922278070924416, −2.56952623045599945109306159820, −1.27175519860935171665972309535,
0.76127933034490834246566132080, 1.41855486644562583620736229463, 2.79951268318519460470700620392, 4.43007153954304543752967691282, 5.84440627258818123837850783820, 6.55311758491542281626033000331, 8.387632824570899955499952086916, 8.886806644754680995105254018856, 10.91691318554004086847139111598, 11.70221931741757287879573753337
