| 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 + 4·13-s + 14-s − 15-s + 16-s + 18-s + 6·19-s − 20-s + 21-s + 2·22-s − 4·23-s + 24-s + 25-s + 4·26-s + 27-s + 28-s − 7·29-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 + 1.10·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s − 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.457092725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.457092725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.09164893935728, −13.92265135992760, −13.48111289933970, −12.82464210800510, −12.27051359235811, −11.80269583864554, −11.42722169876140, −10.81153497875450, −10.37154768598676, −9.626416112072164, −9.023307440097597, −8.712909326692330, −7.881721081459462, −7.561379921066636, −7.096244627377536, −6.280271338197360, −5.835489455048289, −5.248958656997603, −4.507312733607642, −3.816759263840139, −3.713663722154942, −2.909383936397574, −2.173240598169803, −1.442784773476752, −0.7857315650963322,
0.7857315650963322, 1.442784773476752, 2.173240598169803, 2.909383936397574, 3.713663722154942, 3.816759263840139, 4.507312733607642, 5.248958656997603, 5.835489455048289, 6.280271338197360, 7.096244627377536, 7.561379921066636, 7.881721081459462, 8.712909326692330, 9.023307440097597, 9.626416112072164, 10.37154768598676, 10.81153497875450, 11.42722169876140, 11.80269583864554, 12.27051359235811, 12.82464210800510, 13.48111289933970, 13.92265135992760, 14.09164893935728
