| L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·6-s − 2·7-s + 3·8-s + 9-s − 2·12-s − 6·13-s + 2·14-s − 16-s + 17-s − 18-s − 4·19-s − 4·21-s + 4·23-s + 6·24-s − 5·25-s + 6·26-s − 4·27-s + 2·28-s − 4·29-s − 5·32-s − 34-s − 36-s + 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.577·12-s − 1.66·13-s + 0.534·14-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.872·21-s + 0.834·23-s + 1.22·24-s − 25-s + 1.17·26-s − 0.769·27-s + 0.377·28-s − 0.742·29-s − 0.883·32-s − 0.171·34-s − 1/6·36-s + 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.500963500563692620618372759730, −9.255338204943762853762006111844, −8.135651256026089768311803171744, −7.69590295160345676853562099991, −6.66507887004855405016828727435, −5.25707458904248114424974238446, −4.18894653320208233222063937328, −3.11825330034540641784416797750, −2.02912890024341608869485891410, 0,
2.02912890024341608869485891410, 3.11825330034540641784416797750, 4.18894653320208233222063937328, 5.25707458904248114424974238446, 6.66507887004855405016828727435, 7.69590295160345676853562099991, 8.135651256026089768311803171744, 9.255338204943762853762006111844, 9.500963500563692620618372759730
