| L(s) = 1 | − 2.87·2-s + 1.31·3-s − 23.7·4-s + 25·5-s − 3.77·6-s − 104.·7-s + 160.·8-s − 241.·9-s − 71.8·10-s + 121·11-s − 31.1·12-s − 700.·13-s + 301.·14-s + 32.8·15-s + 299.·16-s − 2.23e3·17-s + 693.·18-s + 361·19-s − 593.·20-s − 137.·21-s − 347.·22-s + 1.36e3·23-s + 210.·24-s + 625·25-s + 2.01e3·26-s − 636.·27-s + 2.49e3·28-s + ⋯ |
| L(s) = 1 | − 0.508·2-s + 0.0842·3-s − 0.741·4-s + 0.447·5-s − 0.0428·6-s − 0.809·7-s + 0.885·8-s − 0.992·9-s − 0.227·10-s + 0.301·11-s − 0.0624·12-s − 1.14·13-s + 0.411·14-s + 0.0376·15-s + 0.292·16-s − 1.87·17-s + 0.504·18-s + 0.229·19-s − 0.331·20-s − 0.0682·21-s − 0.153·22-s + 0.536·23-s + 0.0745·24-s + 0.200·25-s + 0.584·26-s − 0.167·27-s + 0.600·28-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
| \(L(3)\) |
\(\approx\) |
\(0.06392329311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06392329311\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.87T + 32T^{2} \) |
| 3 | \( 1 - 1.31T + 243T^{2} \) |
| 7 | \( 1 + 104.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 700.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.23e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.36e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.27e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.79e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.06e4T + 8.58e9T^{2} \) |
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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
−9.116871878348741082148231566293, −8.772240438517583206755536240596, −7.55444354244015448334879723750, −6.79927871745121020924160882612, −5.73217868463421688841836330377, −4.93896875243903479606698511990, −3.89179222778879033700672908517, −2.78642550904122744917178692284, −1.74477024785386074335335994663, −0.11345119227305630415729483211,
0.11345119227305630415729483211, 1.74477024785386074335335994663, 2.78642550904122744917178692284, 3.89179222778879033700672908517, 4.93896875243903479606698511990, 5.73217868463421688841836330377, 6.79927871745121020924160882612, 7.55444354244015448334879723750, 8.772240438517583206755536240596, 9.116871878348741082148231566293
