| L(s) = 1 | − 0.788·2-s + 3-s − 1.37·4-s − 0.788·6-s − 7-s + 2.66·8-s + 9-s + 11-s − 1.37·12-s − 4.93·13-s + 0.788·14-s + 0.652·16-s − 2.30·17-s − 0.788·18-s + 6.49·19-s − 21-s − 0.788·22-s + 2.76·23-s + 2.66·24-s + 3.89·26-s + 27-s + 1.37·28-s + 0.180·29-s − 3.81·31-s − 5.84·32-s + 33-s + 1.81·34-s + ⋯ |
| L(s) = 1 | − 0.557·2-s + 0.577·3-s − 0.688·4-s − 0.322·6-s − 0.377·7-s + 0.942·8-s + 0.333·9-s + 0.301·11-s − 0.397·12-s − 1.36·13-s + 0.210·14-s + 0.163·16-s − 0.559·17-s − 0.185·18-s + 1.49·19-s − 0.218·21-s − 0.168·22-s + 0.577·23-s + 0.543·24-s + 0.763·26-s + 0.192·27-s + 0.260·28-s + 0.0334·29-s − 0.684·31-s − 1.03·32-s + 0.174·33-s + 0.312·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.788T + 2T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 - 0.180T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 + 5.90T + 41T^{2} \) |
| 43 | \( 1 + 2.42T + 43T^{2} \) |
| 47 | \( 1 - 7.88T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 8.82T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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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
−7.62407697754335412653310735796, −7.40412648564070109598724407621, −6.60508582064244152528399466973, −5.34500722993258226408294725064, −4.93510647727042530373372232507, −3.97373020302549484989621265288, −3.25940445872498645512227128484, −2.29058400490187993712726742467, −1.21602655709003799528017336266, 0,
1.21602655709003799528017336266, 2.29058400490187993712726742467, 3.25940445872498645512227128484, 3.97373020302549484989621265288, 4.93510647727042530373372232507, 5.34500722993258226408294725064, 6.60508582064244152528399466973, 7.40412648564070109598724407621, 7.62407697754335412653310735796
