L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 2·11-s + 12-s + 3·13-s + 14-s + 16-s − 7·17-s − 18-s − 21-s + 2·22-s + 3·23-s − 24-s − 3·26-s + 27-s − 28-s − 3·29-s + 3·31-s − 32-s − 2·33-s + 7·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s − 0.235·18-s − 0.218·21-s + 0.426·22-s + 0.625·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + 0.538·31-s − 0.176·32-s − 0.348·33-s + 1.20·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.96067792965800, −12.01635872914293, −11.87145861965819, −11.20112164175690, −10.74356574225597, −10.53145787060037, −9.936561534212179, −9.391534778106213, −9.091867122147588, −8.661425888687517, −8.150513259044244, −7.925046057837232, −7.142703681472373, −6.823624838286403, −6.440994423360717, −5.834374106003874, −5.341723918084016, −4.519481177023235, −4.313372987846343, −3.479922988666782, −3.077045653091895, −2.552565821243689, −1.992704595192096, −1.464933102468690, −0.6847054098908330, 0,
0.6847054098908330, 1.464933102468690, 1.992704595192096, 2.552565821243689, 3.077045653091895, 3.479922988666782, 4.313372987846343, 4.519481177023235, 5.341723918084016, 5.834374106003874, 6.440994423360717, 6.823624838286403, 7.142703681472373, 7.925046057837232, 8.150513259044244, 8.661425888687517, 9.091867122147588, 9.391534778106213, 9.936561534212179, 10.53145787060037, 10.74356574225597, 11.20112164175690, 11.87145861965819, 12.01635872914293, 12.96067792965800