| L(s) = 1 | − 4·2-s + 13.7·3-s + 16·4-s − 55.1·6-s + 49·7-s − 64·8-s − 53.2·9-s + 50.3·11-s + 220.·12-s − 79.3·13-s − 196·14-s + 256·16-s − 145.·17-s + 212.·18-s − 993.·19-s + 675.·21-s − 201.·22-s + 508.·23-s − 881.·24-s + 317.·26-s − 4.08e3·27-s + 784·28-s − 3.48e3·29-s + 1.54e3·31-s − 1.02e3·32-s + 693.·33-s + 581.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.883·3-s + 0.5·4-s − 0.624·6-s + 0.377·7-s − 0.353·8-s − 0.218·9-s + 0.125·11-s + 0.441·12-s − 0.130·13-s − 0.267·14-s + 0.250·16-s − 0.121·17-s + 0.154·18-s − 0.631·19-s + 0.334·21-s − 0.0886·22-s + 0.200·23-s − 0.312·24-s + 0.0920·26-s − 1.07·27-s + 0.188·28-s − 0.768·29-s + 0.289·31-s − 0.176·32-s + 0.110·33-s + 0.0862·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 13.7T + 243T^{2} \) |
| 11 | \( 1 - 50.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 79.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 145.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 993.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 508.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.72e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 266.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.90e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.64e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.54e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.45e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.14e5T + 8.58e9T^{2} \) |
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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
−10.02946957969860746871554294162, −9.067314704831133062362139735974, −8.455710215287439833498373523310, −7.64372096954620399097940626343, −6.59867476533970152355486963791, −5.31731174942797126796885713625, −3.83635028420286544276062956129, −2.66040923522346258716907509602, −1.62726277951026715129555352840, 0,
1.62726277951026715129555352840, 2.66040923522346258716907509602, 3.83635028420286544276062956129, 5.31731174942797126796885713625, 6.59867476533970152355486963791, 7.64372096954620399097940626343, 8.455710215287439833498373523310, 9.067314704831133062362139735974, 10.02946957969860746871554294162
