| L(s) = 1 | + 0.0663·2-s − 0.255·3-s − 1.99·4-s + 5-s − 0.0169·6-s + 2.56·7-s − 0.265·8-s − 2.93·9-s + 0.0663·10-s + 11-s + 0.509·12-s − 6.37·13-s + 0.170·14-s − 0.255·15-s + 3.97·16-s − 0.997·17-s − 0.194·18-s − 19-s − 1.99·20-s − 0.653·21-s + 0.0663·22-s + 4.24·23-s + 0.0676·24-s + 25-s − 0.423·26-s + 1.51·27-s − 5.10·28-s + ⋯ |
| L(s) = 1 | + 0.0469·2-s − 0.147·3-s − 0.997·4-s + 0.447·5-s − 0.00691·6-s + 0.967·7-s − 0.0937·8-s − 0.978·9-s + 0.0209·10-s + 0.301·11-s + 0.147·12-s − 1.76·13-s + 0.0454·14-s − 0.0658·15-s + 0.993·16-s − 0.241·17-s − 0.0459·18-s − 0.229·19-s − 0.446·20-s − 0.142·21-s + 0.0141·22-s + 0.885·23-s + 0.0138·24-s + 0.200·25-s − 0.0830·26-s + 0.291·27-s − 0.965·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0663T + 2T^{2} \) |
| 3 | \( 1 + 0.255T + 3T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 + 0.997T + 17T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 0.460T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 0.845T + 43T^{2} \) |
| 47 | \( 1 - 9.94T + 47T^{2} \) |
| 53 | \( 1 - 0.244T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 7.48T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 3.59T + 71T^{2} \) |
| 73 | \( 1 + 1.87T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−9.239020549987112858070789554719, −8.939613442139701392396292538937, −7.904857824443833951698007109783, −7.11344310019731216419521972083, −5.74624265101396426056730258914, −5.14785368042438691631208502674, −4.46088596250837431256162930448, −3.11689789548599625140381251256, −1.81770516300025504301343018928, 0,
1.81770516300025504301343018928, 3.11689789548599625140381251256, 4.46088596250837431256162930448, 5.14785368042438691631208502674, 5.74624265101396426056730258914, 7.11344310019731216419521972083, 7.904857824443833951698007109783, 8.939613442139701392396292538937, 9.239020549987112858070789554719
