L(s) = 1 | + 19.1·3-s + (55.2 + 8.66i)5-s − 121. i·7-s + 123.·9-s − 298i·11-s − 485.·13-s + (1.05e3 + 165. i)15-s − 420. i·17-s − 2.57e3i·19-s − 2.32e3i·21-s − 717. i·23-s + (2.97e3 + 956. i)25-s − 2.29e3·27-s − 3.73e3i·29-s − 6.22e3·31-s + ⋯ |
L(s) = 1 | + 1.22·3-s + (0.987 + 0.154i)5-s − 0.937i·7-s + 0.506·9-s − 0.742i·11-s − 0.797·13-s + (1.21 + 0.190i)15-s − 0.353i·17-s − 1.63i·19-s − 1.15i·21-s − 0.282i·23-s + (0.952 + 0.306i)25-s − 0.606·27-s − 0.824i·29-s − 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.316659840\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.316659840\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-55.2 - 8.66i)T \) |
good | 3 | \( 1 - 19.1T + 243T^{2} \) |
| 7 | \( 1 + 121. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 298iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 485.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 420. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.57e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 717. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.73e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.91e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.28e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.63e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 4.87e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 8.39e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 420. iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.06e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.80e5iT - 8.58e9T^{2} \) |
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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
−10.45965502202216185676783158561, −9.395797792137018107239071156176, −8.953434145220363689328340917208, −7.68639869443675031890466075771, −6.94035889141567355180883906464, −5.63656289124307296610401937084, −4.32614870012673123857644889375, −3.01759783965403678094677442767, −2.24840760247956606389440022708, −0.67476676310681496254354079087,
1.76831791927789783016103672179, 2.36706606272904450104723339505, 3.60573693998182465514884394674, 5.12664824199661138020422413285, 6.02931760482895390786156342259, 7.39628913609806333830158456538, 8.323146710167734783241895210654, 9.292112446204157633397863598654, 9.651584132717026155778158852722, 10.79452282952754876611579202862