| L(s) = 1 | − 2-s − 2·3-s − 4-s − 5-s + 2·6-s − 4·7-s + 3·8-s + 9-s + 10-s + 2·11-s + 2·12-s − 13-s + 4·14-s + 2·15-s − 16-s + 2·17-s − 18-s − 6·19-s + 20-s + 8·21-s − 2·22-s − 6·23-s − 6·24-s + 25-s + 26-s + 4·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 1.74·21-s − 0.426·22-s − 1.25·23-s − 1.22·24-s + 1/5·25-s + 0.196·26-s + 0.769·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.36541807055163456713423223499, −12.91463619461871953625928127551, −12.13731877933106587879459657638, −10.75277124581317696673490574703, −9.844976583062747102624785032359, −8.679150483937958571354498819427, −7.04231773996815600274218157258, −5.80677556175820100036715333304, −4.03550878858777485512286959394, 0,
4.03550878858777485512286959394, 5.80677556175820100036715333304, 7.04231773996815600274218157258, 8.679150483937958571354498819427, 9.844976583062747102624785032359, 10.75277124581317696673490574703, 12.13731877933106587879459657638, 12.91463619461871953625928127551, 14.36541807055163456713423223499
