L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 11-s + 12-s + 13-s + 2·14-s + 16-s + 17-s + 18-s + 2·21-s + 22-s − 8·23-s + 24-s + 26-s + 27-s + 2·28-s − 4·31-s + 32-s + 33-s + 34-s + 36-s + 4·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.436·21-s + 0.213·22-s − 1.66·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.377·28-s − 0.718·31-s + 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.79081988209565, −12.24085033206686, −11.92671199977134, −11.51477973550199, −10.89858467384981, −10.66117808376894, −10.03090727762196, −9.537975479309633, −9.161987758828759, −8.456048466564262, −8.191194280282144, −7.539461922506119, −7.465604605573983, −6.605615468744650, −6.254276506732056, −5.607984208709819, −5.379417806419874, −4.509105566380114, −4.220751059418050, −3.868533563392850, −3.167083902840616, −2.677629163422240, −2.050914111895517, −1.599811706982525, −1.031614295601465, 0,
1.031614295601465, 1.599811706982525, 2.050914111895517, 2.677629163422240, 3.167083902840616, 3.868533563392850, 4.220751059418050, 4.509105566380114, 5.379417806419874, 5.607984208709819, 6.254276506732056, 6.605615468744650, 7.465604605573983, 7.539461922506119, 8.191194280282144, 8.456048466564262, 9.161987758828759, 9.537975479309633, 10.03090727762196, 10.66117808376894, 10.89858467384981, 11.51477973550199, 11.92671199977134, 12.24085033206686, 12.79081988209565