L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 5·7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 5·13-s − 5·14-s − 15-s + 16-s − 4·17-s + 18-s − 4·19-s + 20-s + 5·21-s − 22-s − 24-s + 25-s − 5·26-s − 27-s − 5·28-s − 2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 1.38·13-s − 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 1.09·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.192·27-s − 0.944·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.63347246572032, −13.08221263676417, −12.76800994494688, −12.36990903761993, −11.97745817509718, −11.42436000280124, −10.70899964389944, −10.37779749572200, −10.03107366961075, −9.378436540864242, −9.204138189001697, −8.400216191633040, −7.688166454640952, −7.039119801913522, −6.833098228909986, −6.264504653650630, −5.922389516591686, −5.435370263833118, −4.672924677823109, −4.378709383360582, −3.755987368533755, −2.949934472857545, −2.614504863640913, −2.147565730621959, −1.170931221075431, 0, 0,
1.170931221075431, 2.147565730621959, 2.614504863640913, 2.949934472857545, 3.755987368533755, 4.378709383360582, 4.672924677823109, 5.435370263833118, 5.922389516591686, 6.264504653650630, 6.833098228909986, 7.039119801913522, 7.688166454640952, 8.400216191633040, 9.204138189001697, 9.378436540864242, 10.03107366961075, 10.37779749572200, 10.70899964389944, 11.42436000280124, 11.97745817509718, 12.36990903761993, 12.76800994494688, 13.08221263676417, 13.63347246572032