| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s − 7-s + 3·8-s + 9-s − 10-s + 12-s − 6·13-s + 14-s − 15-s − 16-s − 18-s + 4·19-s − 20-s + 21-s − 3·24-s + 25-s + 6·26-s − 27-s + 28-s + 2·29-s + 30-s − 5·32-s − 35-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 + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-s − 0.883·32-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9673832101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9673832101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.93855652852538, −14.73727468268986, −13.93398833355242, −13.61038722709455, −12.89162175086413, −12.50854956518765, −11.94641169984595, −11.32976900667891, −10.61928299826993, −10.15619624147581, −9.730826526452549, −9.277194622159613, −8.854023904129364, −7.805562289821918, −7.604716700896733, −6.962943698644965, −6.272476128517237, −5.481670416739031, −5.135130452805737, −4.428573588818876, −3.852885328914742, −2.750774481678863, −2.215351214118109, −1.096616117920754, −0.5270272641305597,
0.5270272641305597, 1.096616117920754, 2.215351214118109, 2.750774481678863, 3.852885328914742, 4.428573588818876, 5.135130452805737, 5.481670416739031, 6.272476128517237, 6.962943698644965, 7.604716700896733, 7.805562289821918, 8.854023904129364, 9.277194622159613, 9.730826526452549, 10.15619624147581, 10.61928299826993, 11.32976900667891, 11.94641169984595, 12.50854956518765, 12.89162175086413, 13.61038722709455, 13.93398833355242, 14.73727468268986, 14.93855652852538
