| L(s) = 1 | + 15.6·3-s + 25·5-s − 49·7-s + 2.28·9-s + 541.·11-s + 762.·13-s + 391.·15-s + 734.·17-s − 1.10e3·19-s − 767.·21-s + 4.52e3·23-s + 625·25-s − 3.76e3·27-s − 4.03e3·29-s − 9.24e3·31-s + 8.48e3·33-s − 1.22e3·35-s + 1.14e4·37-s + 1.19e4·39-s − 1.10e4·41-s + 8.34e3·43-s + 57.2·45-s + 1.06e3·47-s + 2.40e3·49-s + 1.14e4·51-s + 5.21e3·53-s + 1.35e4·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 0.447·5-s − 0.377·7-s + 0.00942·9-s + 1.35·11-s + 1.25·13-s + 0.449·15-s + 0.616·17-s − 0.702·19-s − 0.379·21-s + 1.78·23-s + 0.200·25-s − 0.995·27-s − 0.890·29-s − 1.72·31-s + 1.35·33-s − 0.169·35-s + 1.37·37-s + 1.25·39-s − 1.02·41-s + 0.687·43-s + 0.00421·45-s + 0.0700·47-s + 0.142·49-s + 0.618·51-s + 0.254·53-s + 0.603·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.911031362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.911031362\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 15.6T + 243T^{2} \) |
| 11 | \( 1 - 541.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 762.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 734.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.34e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.21e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.23e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.80e4T + 8.58e9T^{2} \) |
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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
−9.636959377162595669298958514522, −9.044808700169339716050997966405, −8.518922505740379486313802101065, −7.30817625635431393254790418747, −6.39407909179734563265533682086, −5.49256794426873189641786345274, −3.91225471987612405319116144508, −3.32267139364757490387076590674, −2.06961851062914625486818379713, −0.978835996549977126013064815485,
0.978835996549977126013064815485, 2.06961851062914625486818379713, 3.32267139364757490387076590674, 3.91225471987612405319116144508, 5.49256794426873189641786345274, 6.39407909179734563265533682086, 7.30817625635431393254790418747, 8.518922505740379486313802101065, 9.044808700169339716050997966405, 9.636959377162595669298958514522
