L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 3·7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 5·13-s + 3·14-s − 15-s + 16-s + 17-s − 18-s + 7·19-s − 20-s − 3·21-s + 22-s + 7·23-s − 24-s + 25-s + 5·26-s + 27-s − 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.38·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.60·19-s − 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.980·26-s + 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.79287819231573543425504936141, −7.09135281563765877493914816990, −6.85684693522542306609727037754, −5.61144725973219098722702062568, −4.95966490577776588391208669363, −3.79970486252887664147574067429, −3.01422033581051396798336589967, −2.57356475467701799146063712652, −1.17343690260191068297548793425, 0,
1.17343690260191068297548793425, 2.57356475467701799146063712652, 3.01422033581051396798336589967, 3.79970486252887664147574067429, 4.95966490577776588391208669363, 5.61144725973219098722702062568, 6.85684693522542306609727037754, 7.09135281563765877493914816990, 7.79287819231573543425504936141