| L(s) = 1 | + 3-s − 7-s + 9-s − 5·11-s − 13-s − 5·17-s − 21-s + 27-s + 7·29-s + 9·31-s − 5·33-s − 8·37-s − 39-s − 2·41-s − 8·43-s − 9·47-s − 6·49-s − 5·51-s + 11·53-s + 59-s + 7·61-s − 63-s + 15·67-s + 8·71-s − 4·73-s + 5·77-s + 4·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 1.21·17-s − 0.218·21-s + 0.192·27-s + 1.29·29-s + 1.61·31-s − 0.870·33-s − 1.31·37-s − 0.160·39-s − 0.312·41-s − 1.21·43-s − 1.31·47-s − 6/7·49-s − 0.700·51-s + 1.51·53-s + 0.130·59-s + 0.896·61-s − 0.125·63-s + 1.83·67-s + 0.949·71-s − 0.468·73-s + 0.569·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−14.56390874903488, −13.77942296307008, −13.61597709801116, −12.99842569827971, −12.77444429605214, −11.89451988665080, −11.67272606103805, −10.77506903465253, −10.36988555453207, −9.996576139549391, −9.478142420099094, −8.723738983052988, −8.214662550163807, −8.092688170838614, −7.193899532957552, −6.602446730547450, −6.426227067832414, −5.256055362580352, −5.023455889108536, −4.422291419702562, −3.568216583540093, −3.059596999556089, −2.393862402016807, −2.014002567162436, −0.8473084333369638, 0,
0.8473084333369638, 2.014002567162436, 2.393862402016807, 3.059596999556089, 3.568216583540093, 4.422291419702562, 5.023455889108536, 5.256055362580352, 6.426227067832414, 6.602446730547450, 7.193899532957552, 8.092688170838614, 8.214662550163807, 8.723738983052988, 9.478142420099094, 9.996576139549391, 10.36988555453207, 10.77506903465253, 11.67272606103805, 11.89451988665080, 12.77444429605214, 12.99842569827971, 13.61597709801116, 13.77942296307008, 14.56390874903488
