| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s + 9-s + 10-s + 12-s − 2·13-s + 4·14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s + 4·21-s + 24-s + 25-s − 2·26-s + 27-s + 4·28-s + 6·29-s + 30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.711786560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.711786560\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.18874022007591, −13.78317464236013, −13.66559070087424, −12.77780311603154, −12.20927599969465, −11.84393182443587, −11.45371839839350, −10.68270434513970, −10.14142962228229, −9.961550898864896, −8.908339652569029, −8.628457109384491, −8.061488174828707, −7.394644190630757, −7.129797891472181, −6.363935763816548, −5.624845582752910, −5.115605491962071, −4.765034335333403, −4.113869428255549, −3.446715899367254, −2.634130500702864, −2.290675306273740, −1.458567391265456, −0.9224688737259843,
0.9224688737259843, 1.458567391265456, 2.290675306273740, 2.634130500702864, 3.446715899367254, 4.113869428255549, 4.765034335333403, 5.115605491962071, 5.624845582752910, 6.363935763816548, 7.129797891472181, 7.394644190630757, 8.061488174828707, 8.628457109384491, 8.908339652569029, 9.961550898864896, 10.14142962228229, 10.68270434513970, 11.45371839839350, 11.84393182443587, 12.20927599969465, 12.77780311603154, 13.66559070087424, 13.78317464236013, 14.18874022007591
