| L(s) = 1 | + 9·3-s + 44·5-s + 49·7-s + 81·9-s + 470·11-s − 1.15e3·13-s + 396·15-s + 1.20e3·17-s + 2.64e3·19-s + 441·21-s + 1.19e3·23-s − 1.18e3·25-s + 729·27-s + 3.61e3·29-s − 5.61e3·31-s + 4.23e3·33-s + 2.15e3·35-s − 6.47e3·37-s − 1.04e4·39-s + 2.85e3·41-s + 1.34e4·43-s + 3.56e3·45-s + 1.83e4·47-s + 2.40e3·49-s + 1.08e4·51-s − 4.37e3·53-s + 2.06e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.787·5-s + 0.377·7-s + 1/3·9-s + 1.17·11-s − 1.90·13-s + 0.454·15-s + 1.01·17-s + 1.68·19-s + 0.218·21-s + 0.469·23-s − 0.380·25-s + 0.192·27-s + 0.797·29-s − 1.04·31-s + 0.676·33-s + 0.297·35-s − 0.777·37-s − 1.09·39-s + 0.265·41-s + 1.11·43-s + 0.262·45-s + 1.21·47-s + 1/7·49-s + 0.583·51-s − 0.213·53-s + 0.921·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.486073379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.486073379\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 44 T + p^{5} T^{2} \) |
| 11 | \( 1 - 470 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1158 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1204 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2644 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1190 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3614 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5616 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6478 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2856 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13492 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18372 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4374 T + p^{5} T^{2} \) |
| 59 | \( 1 + 30248 T + p^{5} T^{2} \) |
| 61 | \( 1 - 19542 T + p^{5} T^{2} \) |
| 67 | \( 1 + 54328 T + p^{5} T^{2} \) |
| 71 | \( 1 - 10730 T + p^{5} T^{2} \) |
| 73 | \( 1 - 35374 T + p^{5} T^{2} \) |
| 79 | \( 1 - 49956 T + p^{5} T^{2} \) |
| 83 | \( 1 - 26948 T + p^{5} T^{2} \) |
| 89 | \( 1 - 100776 T + p^{5} T^{2} \) |
| 97 | \( 1 - 77134 T + p^{5} T^{2} \) |
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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
−10.49449718003727153147857571412, −9.528814792860584702582707100674, −9.222541769176063587679386635206, −7.72809965360369619706046320597, −7.11880260253048227820804011965, −5.72686433920328827070331187653, −4.79640813849186246757055737601, −3.39853138773885273165666223698, −2.22065175285589959447520579033, −1.07639347891691117862735437866,
1.07639347891691117862735437866, 2.22065175285589959447520579033, 3.39853138773885273165666223698, 4.79640813849186246757055737601, 5.72686433920328827070331187653, 7.11880260253048227820804011965, 7.72809965360369619706046320597, 9.222541769176063587679386635206, 9.528814792860584702582707100674, 10.49449718003727153147857571412
