L(s) = 1 | + 3-s + 7-s − 2·9-s + 3·11-s − 5·13-s − 3·17-s − 2·19-s + 21-s − 6·23-s − 5·27-s + 3·29-s + 4·31-s + 3·33-s − 2·37-s − 5·39-s − 12·41-s − 10·43-s + 9·47-s + 49-s − 3·51-s − 12·53-s − 2·57-s + 8·61-s − 2·63-s − 4·67-s − 6·69-s − 2·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.38·13-s − 0.727·17-s − 0.458·19-s + 0.218·21-s − 1.25·23-s − 0.962·27-s + 0.557·29-s + 0.718·31-s + 0.522·33-s − 0.328·37-s − 0.800·39-s − 1.87·41-s − 1.52·43-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s − 0.264·57-s + 1.02·61-s − 0.251·63-s − 0.488·67-s − 0.722·69-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.430448333394137094980342603890, −7.86011514048466165442018696861, −6.88150838864812305101444716237, −6.27248387518076710770064262455, −5.19532839461654101726843493087, −4.46535279179624622242920849885, −3.55022395369242424556181791228, −2.54455347664810529993661521980, −1.77645134008769781761461177809, 0,
1.77645134008769781761461177809, 2.54455347664810529993661521980, 3.55022395369242424556181791228, 4.46535279179624622242920849885, 5.19532839461654101726843493087, 6.27248387518076710770064262455, 6.88150838864812305101444716237, 7.86011514048466165442018696861, 8.430448333394137094980342603890