L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s + 5·11-s − 12-s − 2·13-s + 2·14-s + 15-s + 16-s − 18-s − 19-s − 20-s + 2·21-s − 5·22-s − 3·23-s + 24-s + 25-s + 2·26-s − 27-s − 2·28-s − 6·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.436·21-s − 1.06·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.49463337576218, −12.70400674219403, −12.36612012024917, −12.12938033589645, −11.52561486656646, −11.13363551653928, −10.60323653232382, −10.22276534231547, −9.464299721889610, −9.253531459587258, −8.962843403183340, −8.114735363738617, −7.652453804005541, −7.079303145962665, −6.837901273303404, −6.145093161236062, −5.852957803250816, −5.204072073695128, −4.361482914433429, −3.955610409493926, −3.503390623389885, −2.739628978658621, −2.008811482478244, −1.415921961268619, −0.6382265422607918, 0,
0.6382265422607918, 1.415921961268619, 2.008811482478244, 2.739628978658621, 3.503390623389885, 3.955610409493926, 4.361482914433429, 5.204072073695128, 5.852957803250816, 6.145093161236062, 6.837901273303404, 7.079303145962665, 7.652453804005541, 8.114735363738617, 8.962843403183340, 9.253531459587258, 9.464299721889610, 10.22276534231547, 10.60323653232382, 11.13363551653928, 11.52561486656646, 12.12938033589645, 12.36612012024917, 12.70400674219403, 13.49463337576218