L(s) = 1 | + 9.80·2-s + 0.501·3-s + 64.0·4-s + 25·5-s + 4.91·6-s + 0.788·7-s + 314.·8-s − 242.·9-s + 245.·10-s − 121·11-s + 32.0·12-s + 446.·13-s + 7.72·14-s + 12.5·15-s + 1.02e3·16-s − 1.92e3·17-s − 2.37e3·18-s + 361·19-s + 1.60e3·20-s + 0.394·21-s − 1.18e3·22-s − 4.28e3·23-s + 157.·24-s + 625·25-s + 4.37e3·26-s − 243.·27-s + 50.4·28-s + ⋯ |
L(s) = 1 | + 1.73·2-s + 0.0321·3-s + 2.00·4-s + 0.447·5-s + 0.0556·6-s + 0.00607·7-s + 1.73·8-s − 0.998·9-s + 0.774·10-s − 0.301·11-s + 0.0643·12-s + 0.732·13-s + 0.0105·14-s + 0.0143·15-s + 1.00·16-s − 1.61·17-s − 1.73·18-s + 0.229·19-s + 0.895·20-s + 0.000195·21-s − 0.522·22-s − 1.68·23-s + 0.0557·24-s + 0.200·25-s + 1.26·26-s − 0.0642·27-s + 0.0121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 - 9.80T + 32T^{2} \) |
| 3 | \( 1 - 0.501T + 243T^{2} \) |
| 7 | \( 1 - 0.788T + 1.68e4T^{2} \) |
| 13 | \( 1 - 446.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.92e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.92e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.40e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.57e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.60e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 316.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.02e5T + 8.58e9T^{2} \) |
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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
−8.670769719661054061842751478813, −7.78374717660934341524037463704, −6.47537554176977792652145628170, −6.14910577150341332252978365403, −5.29140515720964519290302797663, −4.42834636833286654399539388244, −3.53725954319135489804833423853, −2.59217931005556198082712507650, −1.83070993479310807718598916207, 0,
1.83070993479310807718598916207, 2.59217931005556198082712507650, 3.53725954319135489804833423853, 4.42834636833286654399539388244, 5.29140515720964519290302797663, 6.14910577150341332252978365403, 6.47537554176977792652145628170, 7.78374717660934341524037463704, 8.670769719661054061842751478813