L(s) = 1 | − 3.71·2-s + 7.61·3-s + 5.82·4-s − 15.9·5-s − 28.3·6-s + 21.9·7-s + 8.07·8-s + 31.0·9-s + 59.4·10-s + 11·11-s + 44.3·12-s − 81.6·14-s − 121.·15-s − 76.6·16-s − 94.2·17-s − 115.·18-s − 13.8·19-s − 93.1·20-s + 167.·21-s − 40.9·22-s + 77.7·23-s + 61.5·24-s + 130.·25-s + 30.6·27-s + 127.·28-s + 295.·29-s + 452.·30-s + ⋯ |
L(s) = 1 | − 1.31·2-s + 1.46·3-s + 0.728·4-s − 1.42·5-s − 1.92·6-s + 1.18·7-s + 0.356·8-s + 1.14·9-s + 1.87·10-s + 0.301·11-s + 1.06·12-s − 1.55·14-s − 2.09·15-s − 1.19·16-s − 1.34·17-s − 1.51·18-s − 0.167·19-s − 1.04·20-s + 1.73·21-s − 0.396·22-s + 0.704·23-s + 0.523·24-s + 1.04·25-s + 0.218·27-s + 0.863·28-s + 1.89·29-s + 2.75·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 3.71T + 8T^{2} \) |
| 3 | \( 1 - 7.61T + 27T^{2} \) |
| 5 | \( 1 + 15.9T + 125T^{2} \) |
| 7 | \( 1 - 21.9T + 343T^{2} \) |
| 17 | \( 1 + 94.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 145.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 287.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 413.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 522.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 215.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 862.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 281.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 768.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 767.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 641.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 193.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 758.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 994.T + 9.12e5T^{2} \) |
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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
−8.354217875650718277288745879391, −8.231565611720714848708677188749, −7.32268129927687243637270474175, −6.79791192590110857412951869125, −4.76586324250779600846861474934, −4.30314886698882745434865681892, −3.25448833741134825912851955746, −2.17151005777329305267960600670, −1.26199295167186834771247081838, 0,
1.26199295167186834771247081838, 2.17151005777329305267960600670, 3.25448833741134825912851955746, 4.30314886698882745434865681892, 4.76586324250779600846861474934, 6.79791192590110857412951869125, 7.32268129927687243637270474175, 8.231565611720714848708677188749, 8.354217875650718277288745879391