L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 2·11-s − 12-s + 3·13-s + 14-s + 15-s + 16-s + 4·17-s − 18-s − 20-s + 21-s − 2·22-s − 6·23-s + 24-s + 25-s − 3·26-s − 27-s − 28-s + 10·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.192·27-s − 0.188·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.128352673\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128352673\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.48149095128845, −16.14825185469101, −15.62800392601227, −14.97461803799862, −14.27783806102610, −13.70909004972800, −12.92696171929070, −12.23254642169094, −11.85269218342290, −11.37399997942315, −10.60885358459513, −10.08868003334767, −9.632678456349248, −8.798303948139290, −8.186000151554346, −7.748484047761635, −6.732367810849687, −6.479639588638896, −5.748554351471988, −4.951714637135148, −3.964820620781226, −3.490540152562773, −2.454844044767427, −1.344850939526273, −0.6334608580750001,
0.6334608580750001, 1.344850939526273, 2.454844044767427, 3.490540152562773, 3.964820620781226, 4.951714637135148, 5.748554351471988, 6.479639588638896, 6.732367810849687, 7.748484047761635, 8.186000151554346, 8.798303948139290, 9.632678456349248, 10.08868003334767, 10.60885358459513, 11.37399997942315, 11.85269218342290, 12.23254642169094, 12.92696171929070, 13.70909004972800, 14.27783806102610, 14.97461803799862, 15.62800392601227, 16.14825185469101, 16.48149095128845