L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 3·11-s − 12-s + 13-s + 14-s + 16-s + 18-s − 6·19-s − 21-s − 3·22-s − 24-s + 26-s − 27-s + 28-s + 2·29-s − 31-s + 32-s + 3·33-s + 36-s − 11·37-s − 6·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.218·21-s − 0.639·22-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.179·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s − 1.80·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.106358770234513391810143258453, −6.79902601145522781045790304722, −6.63286495389347826056364995693, −5.48148188039544436216935040348, −5.14046460746152441805641349263, −4.29991734501479597297220458969, −3.49634217108107969589020868867, −2.43543548980928024287043193804, −1.55489416600218678215821316645, 0,
1.55489416600218678215821316645, 2.43543548980928024287043193804, 3.49634217108107969589020868867, 4.29991734501479597297220458969, 5.14046460746152441805641349263, 5.48148188039544436216935040348, 6.63286495389347826056364995693, 6.79902601145522781045790304722, 8.106358770234513391810143258453