| L(s) = 1 | + (15.9 − 0.219i)2-s + (38.8 + 12.6i)3-s + (255. − 7.03i)4-s + (209. − 588. i)5-s + (623. + 193. i)6-s + 437. i·7-s + (4.09e3 − 168. i)8-s + (−3.96e3 − 2.87e3i)9-s + (3.22e3 − 9.46e3i)10-s + (−1.40e4 − 1.93e4i)11-s + (1.00e4 + 2.95e3i)12-s + (−4.27e4 − 3.10e4i)13-s + (96.3 + 7.00e3i)14-s + (1.55e4 − 2.01e4i)15-s + (6.54e4 − 3.60e3i)16-s + (7.02e3 + 2.16e4i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0137i)2-s + (0.479 + 0.155i)3-s + (0.999 − 0.0274i)4-s + (0.335 − 0.941i)5-s + (0.481 + 0.149i)6-s + 0.182i·7-s + (0.999 − 0.0412i)8-s + (−0.603 − 0.438i)9-s + (0.322 − 0.946i)10-s + (−0.959 − 1.32i)11-s + (0.483 + 0.142i)12-s + (−1.49 − 1.08i)13-s + (0.00250 + 0.182i)14-s + (0.307 − 0.399i)15-s + (0.998 − 0.0549i)16-s + (0.0841 + 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.39136 - 2.97686i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.39136 - 2.97686i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.9 + 0.219i)T \) |
| 5 | \( 1 + (-209. + 588. i)T \) |
| good | 3 | \( 1 + (-38.8 - 12.6i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 437. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (1.40e4 + 1.93e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (4.27e4 + 3.10e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-7.02e3 - 2.16e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-8.12e4 + 2.64e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-1.96e5 - 2.70e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (1.56e5 - 4.81e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (7.65e5 - 2.48e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.16e6 - 8.49e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-1.65e6 - 1.19e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 5.16e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-4.59e6 - 1.49e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-2.19e6 + 6.75e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-3.48e6 + 4.79e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.86e7 + 1.35e7i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-2.57e6 + 8.35e5i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (4.57e6 + 1.48e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.30e7 - 9.46e6i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (7.05e6 + 2.29e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-3.72e7 + 1.21e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (5.51e7 - 4.00e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-3.82e6 + 1.17e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.27635657616659228831202623216, −11.12786168641085005855973056990, −9.840705150450661420799332783344, −8.578712671208915227676420674386, −7.51721947075920099868328432101, −5.62242887622876589568138809530, −5.24512779207552452545472075184, −3.44499747934501322701213988137, −2.50214365891323435028935683984, −0.64704288302072247828721355919,
2.23406814252636849788529372122, 2.63538695468959499108442448922, 4.38972186782011571573541374229, 5.56007675654501400074671815061, 7.14349972343240140756076361379, 7.52742918504799168318261157532, 9.578318790699861426573995255110, 10.58387503782333076614042312505, 11.66167539297729167867350364474, 12.75024163220212318621596916161
