L(s) = 1 | − 3.41·2-s − 9.99·3-s + 3.64·4-s + 5·5-s + 34.1·6-s − 3.40·7-s + 14.8·8-s + 72.9·9-s − 17.0·10-s + 11·11-s − 36.4·12-s − 6.49·13-s + 11.6·14-s − 49.9·15-s − 79.8·16-s − 39.4·17-s − 248.·18-s + 19·19-s + 18.2·20-s + 34.0·21-s − 37.5·22-s − 88.1·23-s − 148.·24-s + 25·25-s + 22.1·26-s − 459.·27-s − 12.4·28-s + ⋯ |
L(s) = 1 | − 1.20·2-s − 1.92·3-s + 0.455·4-s + 0.447·5-s + 2.32·6-s − 0.184·7-s + 0.657·8-s + 2.70·9-s − 0.539·10-s + 0.301·11-s − 0.876·12-s − 0.138·13-s + 0.222·14-s − 0.860·15-s − 1.24·16-s − 0.562·17-s − 3.25·18-s + 0.229·19-s + 0.203·20-s + 0.354·21-s − 0.363·22-s − 0.799·23-s − 1.26·24-s + 0.200·25-s + 0.167·26-s − 3.27·27-s − 0.0838·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 + 3.41T + 8T^{2} \) |
| 3 | \( 1 + 9.99T + 27T^{2} \) |
| 7 | \( 1 + 3.40T + 343T^{2} \) |
| 13 | \( 1 + 6.49T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 88.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 257.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 46.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 499.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 535.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 167.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 223.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 906.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 215.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 215.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.41T + 5.71e5T^{2} \) |
| 89 | \( 1 - 229.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 517.T + 9.12e5T^{2} \) |
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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
−9.637757091250015736868746513928, −8.346706903428832527666973894614, −7.30300847565850217858023239841, −6.64381161112592384222385879749, −5.86240408369278245131159478082, −4.94803277967408658709739594849, −4.09630535373150732054820505636, −1.95118628005982309430974229897, −0.939224681892638328765909120851, 0,
0.939224681892638328765909120851, 1.95118628005982309430974229897, 4.09630535373150732054820505636, 4.94803277967408658709739594849, 5.86240408369278245131159478082, 6.64381161112592384222385879749, 7.30300847565850217858023239841, 8.346706903428832527666973894614, 9.637757091250015736868746513928