L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s + 9-s + 3·10-s − 11-s − 12-s − 14-s + 3·15-s + 16-s + 7·17-s − 18-s − 19-s − 3·20-s − 21-s + 22-s − 7·23-s + 24-s + 4·25-s − 27-s + 28-s + 3·29-s − 3·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s − 0.218·21-s + 0.213·22-s − 1.45·23-s + 0.204·24-s + 4/5·25-s − 0.192·27-s + 0.188·28-s + 0.557·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6690735170\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6690735170\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.898076037134797332948446484117, −7.58663423389632535617995531918, −6.64452163354514621154107499610, −5.91794309631263039139114586698, −5.16541177539205916739574727700, −4.27520572783648654100867284139, −3.64227404025705936992976709258, −2.68602090632586806212409773589, −1.46853372510881575423801935182, −0.50330060895702483123303804703,
0.50330060895702483123303804703, 1.46853372510881575423801935182, 2.68602090632586806212409773589, 3.64227404025705936992976709258, 4.27520572783648654100867284139, 5.16541177539205916739574727700, 5.91794309631263039139114586698, 6.64452163354514621154107499610, 7.58663423389632535617995531918, 7.898076037134797332948446484117