L(s) = 1 | − 7·13-s + 8·19-s − 5·25-s + 11·31-s − 37-s − 13·43-s − 61-s + 11·67-s − 10·73-s − 13·79-s − 19·97-s − 7·103-s − 19·109-s + ⋯ |
L(s) = 1 | − 1.94·13-s + 1.83·19-s − 25-s + 1.97·31-s − 0.164·37-s − 1.98·43-s − 0.128·61-s + 1.34·67-s − 1.17·73-s − 1.46·79-s − 1.92·97-s − 0.689·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.79369144481287054638523559764, −7.17630898701742428414985429655, −6.52761080763118846523990874094, −5.46490895426067129351911774496, −5.02127912952499085221407770215, −4.19901402137283051885687562260, −3.12641612058767111677049642157, −2.50141338411949579314311527535, −1.34529262368927578434895691229, 0,
1.34529262368927578434895691229, 2.50141338411949579314311527535, 3.12641612058767111677049642157, 4.19901402137283051885687562260, 5.02127912952499085221407770215, 5.46490895426067129351911774496, 6.52761080763118846523990874094, 7.17630898701742428414985429655, 7.79369144481287054638523559764