L(s) = 1 | + (0.707 − 1.22i)3-s + (0.5 + 0.866i)4-s − i·5-s + (−0.499 − 0.866i)9-s + 1.41·12-s + (−0.258 + 0.965i)13-s + (−1.22 − 0.707i)15-s + (−0.499 + 0.866i)16-s + (−0.965 − 1.67i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (0.965 − 1.67i)23-s − 25-s + i·31-s + (0.5 − 0.866i)36-s + (−0.448 − 0.258i)37-s + ⋯ |
L(s) = 1 | + (0.707 − 1.22i)3-s + (0.5 + 0.866i)4-s − i·5-s + (−0.499 − 0.866i)9-s + 1.41·12-s + (−0.258 + 0.965i)13-s + (−1.22 − 0.707i)15-s + (−0.499 + 0.866i)16-s + (−0.965 − 1.67i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (0.965 − 1.67i)23-s − 25-s + i·31-s + (0.5 − 0.866i)36-s + (−0.448 − 0.258i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.611290062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.611290062\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 + (0.258 - 0.965i)T \) |
| 31 | \( 1 - iT \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 0.517T + T^{2} \) |
| 59 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.93iT - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−8.907326762777257643749344359998, −8.448009795033291191074035493334, −7.49001578274797737999214838217, −7.06173686502054042608022955828, −6.45075963905246247102917530870, −4.98311959864351356443514054154, −4.30654970384145470941794363830, −2.85722622951202341065880455604, −2.39521511116765004852760180033, −1.15289720740958487103304703075,
1.80467498983473705864424113214, 2.92170488909230804235853265364, 3.54835996002656772173805227359, 4.53403026673984836331213938128, 5.63332737080760345800047366588, 6.11932303184167126686073946365, 7.28569215041651912289902916673, 7.86330369092436556790776897166, 9.063980510104068636390863966269, 9.571797382088429176130088445520