L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s + 9-s + 3·10-s + 12-s − 5·13-s + 14-s − 3·15-s + 16-s − 18-s − 8·19-s − 3·20-s − 21-s − 23-s − 24-s + 4·25-s + 5·26-s + 27-s − 28-s − 3·29-s + 3·30-s + 2·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s − 1.38·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s − 0.670·20-s − 0.218·21-s − 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.980·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + 0.547·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.90124258899132, −13.32568028594955, −12.59045765366392, −12.33442900463657, −12.04882401556499, −11.34640903025292, −10.81016114569005, −10.45437507202053, −9.867226215776274, −9.372563043797246, −8.868895767321899, −8.391271527079820, −7.852606106274363, −7.639456356837657, −6.987571851279780, −6.553626482494505, −6.040574065617119, −4.926700058020733, −4.712337912280693, −3.971590151877353, −3.382692237120581, −2.997445924392166, −2.075166742172502, −1.807158692828441, −0.5076440639193479, 0,
0.5076440639193479, 1.807158692828441, 2.075166742172502, 2.997445924392166, 3.382692237120581, 3.971590151877353, 4.712337912280693, 4.926700058020733, 6.040574065617119, 6.553626482494505, 6.987571851279780, 7.639456356837657, 7.852606106274363, 8.391271527079820, 8.868895767321899, 9.372563043797246, 9.867226215776274, 10.45437507202053, 10.81016114569005, 11.34640903025292, 12.04882401556499, 12.33442900463657, 12.59045765366392, 13.32568028594955, 13.90124258899132