| L(s) = 1 | − 20.1·3-s + 25·5-s + 49·7-s + 162.·9-s − 427.·11-s − 646.·13-s − 503.·15-s − 1.02e3·17-s + 2.25e3·19-s − 986.·21-s + 2.59e3·23-s + 625·25-s + 1.62e3·27-s + 869.·29-s + 4.40e3·31-s + 8.59e3·33-s + 1.22e3·35-s + 2.55e3·37-s + 1.30e4·39-s + 6.22e3·41-s + 7.75e3·43-s + 4.05e3·45-s + 1.88e3·47-s + 2.40e3·49-s + 2.07e4·51-s − 1.17e4·53-s − 1.06e4·55-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 0.447·5-s + 0.377·7-s + 0.666·9-s − 1.06·11-s − 1.06·13-s − 0.577·15-s − 0.863·17-s + 1.43·19-s − 0.487·21-s + 1.02·23-s + 0.200·25-s + 0.430·27-s + 0.192·29-s + 0.823·31-s + 1.37·33-s + 0.169·35-s + 0.306·37-s + 1.37·39-s + 0.578·41-s + 0.639·43-s + 0.298·45-s + 0.124·47-s + 0.142·49-s + 1.11·51-s − 0.573·53-s − 0.475·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.074286804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074286804\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 20.1T + 243T^{2} \) |
| 11 | \( 1 + 427.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 646.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.25e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 869.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.22e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.75e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.21e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.17e4T + 8.58e9T^{2} \) |
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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
−12.11159598582477571529249108955, −11.20921569538552722389922099237, −10.41707015427650963389091043908, −9.347009442680675662037073709182, −7.78948134162268314520418813995, −6.67397800399970079603551464943, −5.40568048479750788843420185290, −4.82714036596842240427396885373, −2.58914842286890635991602355836, −0.72920375490633428883223405250,
0.72920375490633428883223405250, 2.58914842286890635991602355836, 4.82714036596842240427396885373, 5.40568048479750788843420185290, 6.67397800399970079603551464943, 7.78948134162268314520418813995, 9.347009442680675662037073709182, 10.41707015427650963389091043908, 11.20921569538552722389922099237, 12.11159598582477571529249108955
