L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.190 − 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.190 + 0.587i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.5 + 0.363i)9-s + (0.309 − 0.951i)10-s + (−0.5 − 0.363i)12-s + (−0.5 − 0.363i)13-s + (−0.809 + 0.587i)14-s + (0.190 + 0.587i)15-s + (−0.809 − 0.587i)16-s − 0.618·18-s + (0.618 + 1.90i)19-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.190 − 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.190 + 0.587i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.5 + 0.363i)9-s + (0.309 − 0.951i)10-s + (−0.5 − 0.363i)12-s + (−0.5 − 0.363i)13-s + (−0.809 + 0.587i)14-s + (0.190 + 0.587i)15-s + (−0.809 − 0.587i)16-s − 0.618·18-s + (0.618 + 1.90i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7892200691\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7892200691\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−10.13261710241108002796187596253, −8.685365604817043365893925243870, −7.950346777838671459110445768051, −7.64634354895327450760701874037, −7.00123232752279273750761110073, −5.92823639396485152926069495388, −5.01154292528578222179657269558, −3.90464431133248096813380996188, −2.38912255788269706732813757278, −1.36242121848083035217948115954,
0.999118085833443317557835040990, 2.40163299793942516562535716191, 3.63511377884825734187238321510, 4.44251355265355513159105048187, 5.04277213925155067510430901347, 6.85024891445701741431447899067, 7.43337401280166562826523601348, 8.314450538860507623826729579390, 8.939467213377289921306645618161, 9.541380508284366551852274664088