L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 2·11-s + 12-s − 7·13-s + 14-s + 16-s + 7·17-s − 18-s + 8·19-s − 21-s + 2·22-s + 5·23-s − 24-s + 7·26-s + 27-s − 28-s + 9·29-s + 31-s − 32-s − 2·33-s − 7·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.94·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.83·19-s − 0.218·21-s + 0.426·22-s + 1.04·23-s − 0.204·24-s + 1.37·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s + 0.179·31-s − 0.176·32-s − 0.348·33-s − 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.339863646\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.339863646\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 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
−9.838791532098356959217399335575, −9.314276201894361189726268747264, −8.132788049862507970445322945220, −7.54392364458419480987676267402, −6.97587151947823251519146924106, −5.59816972376066845528951357350, −4.79565646870238517074069242438, −3.14773582507890044917606243068, −2.66687668466195404295404364048, −0.989278977507431980256878998913,
0.989278977507431980256878998913, 2.66687668466195404295404364048, 3.14773582507890044917606243068, 4.79565646870238517074069242438, 5.59816972376066845528951357350, 6.97587151947823251519146924106, 7.54392364458419480987676267402, 8.132788049862507970445322945220, 9.314276201894361189726268747264, 9.838791532098356959217399335575