| L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s + 2·11-s − 2·15-s − 3·17-s − 8·19-s − 2·21-s − 6·23-s − 4·25-s − 4·27-s + 9·29-s + 6·31-s + 4·33-s + 35-s − 3·37-s − 3·41-s − 2·43-s − 45-s + 12·47-s + 49-s − 6·51-s + 5·53-s − 2·55-s − 16·57-s − 14·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.516·15-s − 0.727·17-s − 1.83·19-s − 0.436·21-s − 1.25·23-s − 4/5·25-s − 0.769·27-s + 1.67·29-s + 1.07·31-s + 0.696·33-s + 0.169·35-s − 0.493·37-s − 0.468·41-s − 0.304·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.686·53-s − 0.269·55-s − 2.11·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707169496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707169496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.94344954525165, −13.59905337766463, −13.36827441648051, −12.45849422777098, −12.05849451938240, −11.82958469607136, −10.89723741554783, −10.53240855079551, −9.958876352263256, −9.433275655591649, −8.770603975800323, −8.559981416523071, −8.060773532852062, −7.582167747685383, −6.762858338178291, −6.397012920065509, −5.950295828125369, −4.977497946818900, −4.200716034434845, −4.032888221708792, −3.413002923849221, −2.500980014296367, −2.345333109100461, −1.480997769554114, −0.3798279828336423,
0.3798279828336423, 1.480997769554114, 2.345333109100461, 2.500980014296367, 3.413002923849221, 4.032888221708792, 4.200716034434845, 4.977497946818900, 5.950295828125369, 6.397012920065509, 6.762858338178291, 7.582167747685383, 8.060773532852062, 8.559981416523071, 8.770603975800323, 9.433275655591649, 9.958876352263256, 10.53240855079551, 10.89723741554783, 11.82958469607136, 12.05849451938240, 12.45849422777098, 13.36827441648051, 13.59905337766463, 13.94344954525165
