L(s) = 1 | + 2-s + 1.82·3-s + 4-s + 5-s + 1.82·6-s − 3·7-s + 8-s + 0.318·9-s + 10-s − 11-s + 1.82·12-s − 1.45·13-s − 3·14-s + 1.82·15-s + 16-s − 2.70·17-s + 0.318·18-s − 8.40·19-s + 20-s − 5.46·21-s − 22-s + 8.44·23-s + 1.82·24-s + 25-s − 1.45·26-s − 4.88·27-s − 3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s + 0.447·5-s + 0.743·6-s − 1.13·7-s + 0.353·8-s + 0.106·9-s + 0.316·10-s − 0.301·11-s + 0.525·12-s − 0.404·13-s − 0.801·14-s + 0.470·15-s + 0.250·16-s − 0.654·17-s + 0.0750·18-s − 1.92·19-s + 0.223·20-s − 1.19·21-s − 0.213·22-s + 1.75·23-s + 0.371·24-s + 0.200·25-s − 0.286·26-s − 0.940·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 8.40T + 19T^{2} \) |
| 23 | \( 1 - 8.44T + 23T^{2} \) |
| 29 | \( 1 + 0.764T + 29T^{2} \) |
| 31 | \( 1 - 6.91T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 + 2.51T + 41T^{2} \) |
| 43 | \( 1 - 0.955T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 - 3.15T + 59T^{2} \) |
| 61 | \( 1 + 0.272T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 79 | \( 1 + 9.71T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 5.18T + 89T^{2} \) |
| 97 | \( 1 + 5.27T + 97T^{2} \) |
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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
−7.33603137542634248140442201976, −6.64055042180571680814519653490, −6.28676583600724803227376334866, −5.32584035231292581580031468272, −4.56465106598960938582161675675, −3.80392356536836248685368621867, −2.81869688111315920307314155906, −2.72935851969472907526536802035, −1.69275863612017914628669453738, 0,
1.69275863612017914628669453738, 2.72935851969472907526536802035, 2.81869688111315920307314155906, 3.80392356536836248685368621867, 4.56465106598960938582161675675, 5.32584035231292581580031468272, 6.28676583600724803227376334866, 6.64055042180571680814519653490, 7.33603137542634248140442201976