L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s − 2·9-s + 2·10-s + 2·11-s − 12-s + 4·13-s − 2·15-s + 16-s − 2·18-s − 19-s + 2·20-s + 2·22-s − 24-s − 25-s + 4·26-s + 5·27-s + 3·29-s − 2·30-s − 7·31-s + 32-s − 2·33-s − 2·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.632·10-s + 0.603·11-s − 0.288·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.426·22-s − 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.962·27-s + 0.557·29-s − 0.365·30-s − 1.25·31-s + 0.176·32-s − 0.348·33-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.49679610738810, −14.79338781045994, −14.26484985179863, −13.94133267805013, −13.32669147876151, −12.97816220400699, −12.16537518597137, −11.84872409147005, −11.19188300198797, −10.76554523763721, −10.30730136163824, −9.464566085113266, −9.034226468215274, −8.370763535706658, −7.745840342963355, −6.726726984356045, −6.515670648486205, −5.732237540785190, −5.650673911736354, −4.780594552189101, −4.132120720313332, −3.360641852412749, −2.791912716639016, −1.788294863235343, −1.313489896361748, 0,
1.313489896361748, 1.788294863235343, 2.791912716639016, 3.360641852412749, 4.132120720313332, 4.780594552189101, 5.650673911736354, 5.732237540785190, 6.515670648486205, 6.726726984356045, 7.745840342963355, 8.370763535706658, 9.034226468215274, 9.464566085113266, 10.30730136163824, 10.76554523763721, 11.19188300198797, 11.84872409147005, 12.16537518597137, 12.97816220400699, 13.32669147876151, 13.94133267805013, 14.26484985179863, 14.79338781045994, 15.49679610738810