L(s) = 1 | + 3·5-s − 3·9-s − 11-s − 6·17-s − 19-s − 3·23-s + 4·25-s + 6·29-s + 2·31-s + 6·41-s + 9·43-s − 9·45-s − 3·47-s − 2·53-s − 3·55-s − 12·59-s + 11·61-s − 8·67-s + 6·71-s − 15·73-s − 12·79-s + 9·81-s + 7·83-s − 18·85-s − 8·89-s − 3·95-s − 6·97-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 9-s − 0.301·11-s − 1.45·17-s − 0.229·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s + 0.359·31-s + 0.937·41-s + 1.37·43-s − 1.34·45-s − 0.437·47-s − 0.274·53-s − 0.404·55-s − 1.56·59-s + 1.40·61-s − 0.977·67-s + 0.712·71-s − 1.75·73-s − 1.35·79-s + 81-s + 0.768·83-s − 1.95·85-s − 0.847·89-s − 0.307·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.58307544557329539967991634201, −6.56328635986001124398269900992, −6.17853327134394752085293564851, −5.57822082207509710319393482103, −4.80452478437771912924885227405, −4.04009108968113811912972482337, −2.68073936446631275133467515315, −2.50028934371806191442161108537, −1.40771310623096523722321225806, 0,
1.40771310623096523722321225806, 2.50028934371806191442161108537, 2.68073936446631275133467515315, 4.04009108968113811912972482337, 4.80452478437771912924885227405, 5.57822082207509710319393482103, 6.17853327134394752085293564851, 6.56328635986001124398269900992, 7.58307544557329539967991634201