| L(s) = 1 | − 4.35·2-s + 17.4·3-s + 3.00·4-s + 25·5-s − 76.0·6-s + 56.6·8-s + 223.·9-s − 108.·10-s + 62·11-s + 52.3·12-s − 331.·13-s + 435.·15-s − 295·16-s − 972.·18-s + 361·19-s + 75.0·20-s − 270.·22-s + 987.·24-s + 625·25-s + 1.44e3·26-s + 2.47e3·27-s − 1.90e3·30-s + 379.·32-s + 1.08e3·33-s + 669.·36-s + 714.·37-s − 1.57e3·38-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 1.93·3-s + 0.187·4-s + 5-s − 2.11·6-s + 0.885·8-s + 2.75·9-s − 1.08·10-s + 0.512·11-s + 0.363·12-s − 1.96·13-s + 1.93·15-s − 1.15·16-s − 3.00·18-s + 19-s + 0.187·20-s − 0.558·22-s + 1.71·24-s + 25-s + 2.13·26-s + 3.39·27-s − 2.11·30-s + 0.370·32-s + 0.992·33-s + 0.516·36-s + 0.522·37-s − 1.08·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.083015337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.083015337\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 4.35T + 16T^{2} \) |
| 3 | \( 1 - 17.4T + 81T^{2} \) |
| 7 | \( 1 - 2.40e3T^{2} \) |
| 11 | \( 1 - 62T + 1.46e4T^{2} \) |
| 13 | \( 1 + 331.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 8.35e4T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 - 714.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 + 5.56e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 + 7.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.60e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.07e4T + 8.85e7T^{2} \) |
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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
−13.63112429628669257412410397800, −12.48627851320936524858608739493, −10.31308597026365321269873585868, −9.497509198220176929943932610975, −9.196682763793450800802626843346, −7.896901994693737075867863953016, −7.10848190054409813626417636181, −4.64158415850987255923859574541, −2.75774852437567556429450832923, −1.55415972787308553030364750114,
1.55415972787308553030364750114, 2.75774852437567556429450832923, 4.64158415850987255923859574541, 7.10848190054409813626417636181, 7.896901994693737075867863953016, 9.196682763793450800802626843346, 9.497509198220176929943932610975, 10.31308597026365321269873585868, 12.48627851320936524858608739493, 13.63112429628669257412410397800
