L(s) = 1 | + 6.55·2-s − 14.9·3-s + 10.9·4-s − 98.2·6-s + 156.·7-s − 137.·8-s − 18.4·9-s − 121·11-s − 164.·12-s + 895.·13-s + 1.02e3·14-s − 1.25e3·16-s − 1.36e3·17-s − 120.·18-s − 943.·19-s − 2.34e3·21-s − 793.·22-s − 35.0·23-s + 2.06e3·24-s + 5.87e3·26-s + 3.91e3·27-s + 1.72e3·28-s + 8.55e3·29-s + 3.57e3·31-s − 3.82e3·32-s + 1.81e3·33-s − 8.95e3·34-s + ⋯ |
L(s) = 1 | + 1.15·2-s − 0.961·3-s + 0.343·4-s − 1.11·6-s + 1.20·7-s − 0.760·8-s − 0.0759·9-s − 0.301·11-s − 0.330·12-s + 1.47·13-s + 1.40·14-s − 1.22·16-s − 1.14·17-s − 0.0880·18-s − 0.599·19-s − 1.16·21-s − 0.349·22-s − 0.0138·23-s + 0.731·24-s + 1.70·26-s + 1.03·27-s + 0.415·28-s + 1.88·29-s + 0.667·31-s − 0.659·32-s + 0.289·33-s − 1.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.553649559\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.553649559\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 6.55T + 32T^{2} \) |
| 3 | \( 1 + 14.9T + 243T^{2} \) |
| 7 | \( 1 - 156.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 895.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.36e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 943.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 35.0T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.94e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.22e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.21e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.08e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.58e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.32e5T + 8.58e9T^{2} \) |
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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
−11.12252064552110802018345245366, −10.77585440753687359758707868741, −8.953931050103450867002357894473, −8.172105437410844515314736206619, −6.48667069244365192108348024132, −5.89070781100477951055503840681, −4.82532021247059308939357290403, −4.20109851118775083969194636299, −2.57243159165048284674086588362, −0.824017471144216755436612196808,
0.824017471144216755436612196808, 2.57243159165048284674086588362, 4.20109851118775083969194636299, 4.82532021247059308939357290403, 5.89070781100477951055503840681, 6.48667069244365192108348024132, 8.172105437410844515314736206619, 8.953931050103450867002357894473, 10.77585440753687359758707868741, 11.12252064552110802018345245366