| L(s) = 1 | − 2-s + 1.23·3-s + 4-s − 3.23·5-s − 1.23·6-s − 7-s − 8-s − 1.47·9-s + 3.23·10-s + 1.23·12-s − 6.47·13-s + 14-s − 4.00·15-s + 16-s − 2.76·17-s + 1.47·18-s + 4.47·19-s − 3.23·20-s − 1.23·21-s − 23-s − 1.23·24-s + 5.47·25-s + 6.47·26-s − 5.52·27-s − 28-s − 2·29-s + 4.00·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.713·3-s + 0.5·4-s − 1.44·5-s − 0.504·6-s − 0.377·7-s − 0.353·8-s − 0.490·9-s + 1.02·10-s + 0.356·12-s − 1.79·13-s + 0.267·14-s − 1.03·15-s + 0.250·16-s − 0.670·17-s + 0.346·18-s + 1.02·19-s − 0.723·20-s − 0.269·21-s − 0.208·23-s − 0.252·24-s + 1.09·25-s + 1.26·26-s − 1.06·27-s − 0.188·28-s − 0.371·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.76T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−11.33598668041563251144341240242, −9.966100579885103202415326426250, −9.254380197648161154993510965107, −8.161283154031306834787852567113, −7.67330535918957980218888662552, −6.71875768911979054641360893053, −4.99184512609871567995818055417, −3.58094090961027720456063710771, −2.55478921332239202137526928945, 0,
2.55478921332239202137526928945, 3.58094090961027720456063710771, 4.99184512609871567995818055417, 6.71875768911979054641360893053, 7.67330535918957980218888662552, 8.161283154031306834787852567113, 9.254380197648161154993510965107, 9.966100579885103202415326426250, 11.33598668041563251144341240242
