| L(s) = 1 | + 2.98·2-s − 3·3-s + 0.886·4-s − 19.9·5-s − 8.94·6-s − 13.4·7-s − 21.2·8-s + 9·9-s − 59.5·10-s + 11.7·11-s − 2.65·12-s − 32.5·13-s − 39.9·14-s + 59.9·15-s − 70.3·16-s − 16.7·17-s + 26.8·18-s − 45.7·19-s − 17.6·20-s + 40.2·21-s + 35.0·22-s − 73.3·23-s + 63.6·24-s + 273.·25-s − 96.9·26-s − 27·27-s − 11.8·28-s + ⋯ |
| L(s) = 1 | + 1.05·2-s − 0.577·3-s + 0.110·4-s − 1.78·5-s − 0.608·6-s − 0.723·7-s − 0.937·8-s + 0.333·9-s − 1.88·10-s + 0.321·11-s − 0.0639·12-s − 0.694·13-s − 0.762·14-s + 1.03·15-s − 1.09·16-s − 0.238·17-s + 0.351·18-s − 0.552·19-s − 0.197·20-s + 0.417·21-s + 0.339·22-s − 0.665·23-s + 0.541·24-s + 2.18·25-s − 0.731·26-s − 0.192·27-s − 0.0801·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6609899350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6609899350\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 211 | \( 1 + 211T \) |
| good | 2 | \( 1 - 2.98T + 8T^{2} \) |
| 5 | \( 1 + 19.9T + 125T^{2} \) |
| 7 | \( 1 + 13.4T + 343T^{2} \) |
| 11 | \( 1 - 11.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6.16T + 2.43e4T^{2} \) |
| 31 | \( 1 - 37.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 54.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 0.252T + 6.89e4T^{2} \) |
| 43 | \( 1 - 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 559.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 156.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 122.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 366.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 172.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 464.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 184.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 608.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 576.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 29.3T + 9.12e5T^{2} \) |
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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
−10.43791491703213921048597292208, −9.295513549790017480390866958443, −8.347219563673800171596902694887, −7.27684962099711213719501969027, −6.52073053361916850611847368684, −5.45527256881930600016106540486, −4.32733270162900546806745094529, −3.96003183566263444772817115536, −2.82716980760457676709111137470, −0.40042605861474268880077922875,
0.40042605861474268880077922875, 2.82716980760457676709111137470, 3.96003183566263444772817115536, 4.32733270162900546806745094529, 5.45527256881930600016106540486, 6.52073053361916850611847368684, 7.27684962099711213719501969027, 8.347219563673800171596902694887, 9.295513549790017480390866958443, 10.43791491703213921048597292208
