L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 3·7-s + 8-s + 9-s − 10-s + 2·11-s − 12-s + 13-s + 3·14-s + 15-s + 16-s + 18-s − 2·19-s − 20-s − 3·21-s + 2·22-s − 24-s + 25-s + 26-s − 27-s + 3·28-s + 30-s − 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 + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.654·21-s + 0.426·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s + 0.182·30-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.104035149\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.104035149\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.48353864629970, −11.88083532551791, −11.59741970844870, −11.31645377810878, −10.93868041372840, −10.34140642417091, −10.01887609688370, −9.228453281850165, −8.864201181581746, −8.253469704825811, −7.784296165991361, −7.521156703851403, −6.791495528011006, −6.438644194198521, −5.967048365601844, −5.448439084325934, −4.839915719296163, −4.599324725412998, −4.075486775145370, −3.601541769755983, −3.022446967194145, −2.153947346200485, −1.859319317678230, −1.047611573603392, −0.6028295213139608,
0.6028295213139608, 1.047611573603392, 1.859319317678230, 2.153947346200485, 3.022446967194145, 3.601541769755983, 4.075486775145370, 4.599324725412998, 4.839915719296163, 5.448439084325934, 5.967048365601844, 6.438644194198521, 6.791495528011006, 7.521156703851403, 7.784296165991361, 8.253469704825811, 8.864201181581746, 9.228453281850165, 10.01887609688370, 10.34140642417091, 10.93868041372840, 11.31645377810878, 11.59741970844870, 11.88083532551791, 12.48353864629970