| L(s) = 1 | + 7.68·3-s − 18.6·5-s − 10.3·7-s + 32.0·9-s + 17.0·11-s + 27.4·13-s − 143.·15-s + 48.9·17-s − 51.9·19-s − 79.2·21-s + 16.3·23-s + 224.·25-s + 38.4·27-s + 29·29-s + 240.·31-s + 131·33-s + 192.·35-s − 300.·37-s + 210.·39-s − 105.·41-s − 519.·43-s − 598.·45-s + 375.·47-s − 236.·49-s + 376.·51-s − 673.·53-s − 318.·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 1.67·5-s − 0.557·7-s + 1.18·9-s + 0.467·11-s + 0.584·13-s − 2.47·15-s + 0.698·17-s − 0.626·19-s − 0.823·21-s + 0.148·23-s + 1.79·25-s + 0.274·27-s + 0.185·29-s + 1.39·31-s + 0.691·33-s + 0.931·35-s − 1.33·37-s + 0.864·39-s − 0.401·41-s − 1.84·43-s − 1.98·45-s + 1.16·47-s − 0.689·49-s + 1.03·51-s − 1.74·53-s − 0.781·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 7.68T + 27T^{2} \) |
| 5 | \( 1 + 18.6T + 125T^{2} \) |
| 7 | \( 1 + 10.3T + 343T^{2} \) |
| 11 | \( 1 - 17.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.3T + 1.21e4T^{2} \) |
| 31 | \( 1 - 240.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 105.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 519.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 375.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 673.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 195.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 636.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 81.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 137.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 137.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 204.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 471.T + 9.12e5T^{2} \) |
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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
−8.300124599997148834972714802349, −8.059087670027017241873757886024, −7.08595120882517544475511956960, −6.44284123017864585843467394359, −4.91587981326604696165938665622, −3.88540907339890655498037875636, −3.52216882087998389141795343595, −2.75474613666214176659348318947, −1.34935474517169406055107866548, 0,
1.34935474517169406055107866548, 2.75474613666214176659348318947, 3.52216882087998389141795343595, 3.88540907339890655498037875636, 4.91587981326604696165938665622, 6.44284123017864585843467394359, 7.08595120882517544475511956960, 8.059087670027017241873757886024, 8.300124599997148834972714802349
