| L(s) = 1 | + 3·2-s + 3·3-s + 4-s + 9·6-s + 7·7-s − 21·8-s + 9·9-s + 11·11-s + 3·12-s + 16·13-s + 21·14-s − 71·16-s − 21·17-s + 27·18-s + 125·19-s + 21·21-s + 33·22-s + 81·23-s − 63·24-s + 48·26-s + 27·27-s + 7·28-s + 186·29-s − 58·31-s − 45·32-s + 33·33-s − 63·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.612·6-s + 0.377·7-s − 0.928·8-s + 1/3·9-s + 0.301·11-s + 0.0721·12-s + 0.341·13-s + 0.400·14-s − 1.10·16-s − 0.299·17-s + 0.353·18-s + 1.50·19-s + 0.218·21-s + 0.319·22-s + 0.734·23-s − 0.535·24-s + 0.362·26-s + 0.192·27-s + 0.0472·28-s + 1.19·29-s − 0.336·31-s − 0.248·32-s + 0.174·33-s − 0.317·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.425416745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.425416745\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 - p T + p^{3} T^{2} \) |
| 13 | \( 1 - 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 125 T + p^{3} T^{2} \) |
| 23 | \( 1 - 81 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 58 T + p^{3} T^{2} \) |
| 37 | \( 1 - 253 T + p^{3} T^{2} \) |
| 41 | \( 1 - 63 T + p^{3} T^{2} \) |
| 43 | \( 1 - 100 T + p^{3} T^{2} \) |
| 47 | \( 1 - 219 T + p^{3} T^{2} \) |
| 53 | \( 1 - 192 T + p^{3} T^{2} \) |
| 59 | \( 1 - 249 T + p^{3} T^{2} \) |
| 61 | \( 1 + 64 T + p^{3} T^{2} \) |
| 67 | \( 1 + 272 T + p^{3} T^{2} \) |
| 71 | \( 1 + 645 T + p^{3} T^{2} \) |
| 73 | \( 1 - 112 T + p^{3} T^{2} \) |
| 79 | \( 1 - 509 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1254 T + p^{3} T^{2} \) |
| 89 | \( 1 - 756 T + p^{3} T^{2} \) |
| 97 | \( 1 + 839 T + p^{3} T^{2} \) |
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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
−9.639652374998064389237804218289, −9.037720096340605008198025654527, −8.155836786359026785587229140631, −7.16077894071777179025798324460, −6.16937737615898112702890821733, −5.19551255132211431023422064775, −4.40203340799096962969755192618, −3.46347579071509411315482481380, −2.59677707141975815543749120580, −1.03299462517192302958984822054,
1.03299462517192302958984822054, 2.59677707141975815543749120580, 3.46347579071509411315482481380, 4.40203340799096962969755192618, 5.19551255132211431023422064775, 6.16937737615898112702890821733, 7.16077894071777179025798324460, 8.155836786359026785587229140631, 9.037720096340605008198025654527, 9.639652374998064389237804218289
