L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s − 11-s + 12-s − 4·13-s − 2·14-s + 16-s − 2·17-s + 18-s + 8·19-s − 2·21-s − 22-s + 2·23-s + 24-s − 5·25-s − 4·26-s + 27-s − 2·28-s + 2·29-s + 8·31-s + 32-s − 33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.436·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.93138168325349, −13.63230762727429, −13.17697751718090, −12.88040769143233, −12.02205237998431, −11.81120675149799, −11.46291111808346, −10.47947648917406, −9.976964698311359, −9.868805659241990, −9.166419655956386, −8.621450651263937, −7.857510439620235, −7.541832635596587, −7.001600989663615, −6.472747804864588, −5.908541276166104, −5.211562354616293, −4.779792152420853, −4.241774852538048, −3.389726258271705, −3.073024091835415, −2.570350460218261, −1.859273133984862, −1.006414337794552, 0,
1.006414337794552, 1.859273133984862, 2.570350460218261, 3.073024091835415, 3.389726258271705, 4.241774852538048, 4.779792152420853, 5.211562354616293, 5.908541276166104, 6.472747804864588, 7.001600989663615, 7.541832635596587, 7.857510439620235, 8.621450651263937, 9.166419655956386, 9.868805659241990, 9.976964698311359, 10.47947648917406, 11.46291111808346, 11.81120675149799, 12.02205237998431, 12.88040769143233, 13.17697751718090, 13.63230762727429, 13.93138168325349