| L(s) = 1 | + 3.73·5-s + 6.46·13-s + 5.73·17-s + 8.92·25-s − 10.6·29-s − 9.39·37-s + 8·41-s − 7·49-s + 4·53-s + 15.3·61-s + 24.1·65-s − 16.8·73-s + 21.3·85-s − 0.660·89-s − 18·97-s + 20·101-s − 14.3·109-s − 4.12·113-s + ⋯ |
| L(s) = 1 | + 1.66·5-s + 1.79·13-s + 1.39·17-s + 1.78·25-s − 1.97·29-s − 1.54·37-s + 1.24·41-s − 49-s + 0.549·53-s + 1.97·61-s + 2.99·65-s − 1.97·73-s + 2.32·85-s − 0.0699·89-s − 1.82·97-s + 1.99·101-s − 1.37·109-s − 0.387·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.865206093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865206093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 0.660T + 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−8.994210977633164635761506859403, −8.274485677124726335553975472353, −7.27009498010490530718143265202, −6.39474368644208140190466486211, −5.65676368469135902964853806025, −5.40954846759182740913614781329, −3.95727081800436517263191744320, −3.14307793317028215606377158882, −1.93244442607624280792188888914, −1.20945780797206098361271470292,
1.20945780797206098361271470292, 1.93244442607624280792188888914, 3.14307793317028215606377158882, 3.95727081800436517263191744320, 5.40954846759182740913614781329, 5.65676368469135902964853806025, 6.39474368644208140190466486211, 7.27009498010490530718143265202, 8.274485677124726335553975472353, 8.994210977633164635761506859403
