| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 3·5-s + 2·6-s − 5·7-s + 9-s − 6·10-s + 11-s + 2·12-s − 2·13-s − 10·14-s − 3·15-s − 4·16-s − 17-s + 2·18-s − 6·20-s − 5·21-s + 2·22-s − 4·23-s + 4·25-s − 4·26-s + 27-s − 10·28-s + 2·29-s − 6·30-s + 6·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s + 0.816·6-s − 1.88·7-s + 1/3·9-s − 1.89·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s − 2.67·14-s − 0.774·15-s − 16-s − 0.242·17-s + 0.471·18-s − 1.34·20-s − 1.09·21-s + 0.426·22-s − 0.834·23-s + 4/5·25-s − 0.784·26-s + 0.192·27-s − 1.88·28-s + 0.371·29-s − 1.09·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.503827882841026216201178111482, −8.598738845501152801902892747277, −7.58489736631665346168199395971, −6.70211043109012358161209293170, −6.16139530761373947123603078754, −4.78903745519092577014884991606, −3.96503328962323463176155683264, −3.37953587462828366448462440771, −2.64121548040677802901277139388, 0,
2.64121548040677802901277139388, 3.37953587462828366448462440771, 3.96503328962323463176155683264, 4.78903745519092577014884991606, 6.16139530761373947123603078754, 6.70211043109012358161209293170, 7.58489736631665346168199395971, 8.598738845501152801902892747277, 9.503827882841026216201178111482
