| L(s) = 1 | − 3-s + 7-s + 9-s − 2·11-s + 3·13-s + 5·17-s + 4·19-s − 21-s + 3·23-s − 27-s + 29-s + 5·31-s + 2·33-s + 2·37-s − 3·39-s − 41-s + 11·43-s + 8·47-s + 49-s − 5·51-s − 13·53-s − 4·57-s + 3·59-s − 61-s + 63-s + 12·67-s − 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.832·13-s + 1.21·17-s + 0.917·19-s − 0.218·21-s + 0.625·23-s − 0.192·27-s + 0.185·29-s + 0.898·31-s + 0.348·33-s + 0.328·37-s − 0.480·39-s − 0.156·41-s + 1.67·43-s + 1.16·47-s + 1/7·49-s − 0.700·51-s − 1.78·53-s − 0.529·57-s + 0.390·59-s − 0.128·61-s + 0.125·63-s + 1.46·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.325518569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.325518569\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.90719024449174, −15.53239920365148, −14.84380401626383, −14.02205312221964, −13.89261265289471, −13.04219952638513, −12.48717512107139, −12.05996066573845, −11.31206409351596, −10.94418933720298, −10.39261478662620, −9.678080588200503, −9.246868766323933, −8.263709717572759, −7.934321223929057, −7.261304802336659, −6.581326388955157, −5.797997955815662, −5.418522002149795, −4.754870863004547, −3.978441179510255, −3.212589393263581, −2.474227711186243, −1.304796006051071, −0.7756431466026283,
0.7756431466026283, 1.304796006051071, 2.474227711186243, 3.212589393263581, 3.978441179510255, 4.754870863004547, 5.418522002149795, 5.797997955815662, 6.581326388955157, 7.261304802336659, 7.934321223929057, 8.263709717572759, 9.246868766323933, 9.678080588200503, 10.39261478662620, 10.94418933720298, 11.31206409351596, 12.05996066573845, 12.48717512107139, 13.04219952638513, 13.89261265289471, 14.02205312221964, 14.84380401626383, 15.53239920365148, 15.90719024449174
