| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 7-s + 3·8-s + 9-s + 10-s − 12-s + 14-s − 15-s − 16-s + 2·17-s − 18-s + 8·19-s + 20-s − 21-s + 8·23-s + 3·24-s + 25-s + 27-s + 28-s − 2·29-s + 30-s − 4·31-s − 5·32-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 + 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.218·21-s + 1.66·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-s − 0.718·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17745 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17745 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.626229390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626229390\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.00289796795112, −15.25119960970857, −14.72319688510397, −14.20444896516345, −13.58889438949753, −13.15873457343860, −12.59563388731815, −11.97333604893348, −11.16176524211284, −10.78348061196340, −9.939277335287134, −9.507193260939047, −9.113987708376733, −8.558534996573224, −7.761510460664735, −7.464822166483185, −6.960076572555169, −5.904677737200461, −5.131350966148664, −4.664436544680546, −3.613693984917075, −3.376857807915165, −2.400721524717608, −1.255096909134953, −0.6852977581185580,
0.6852977581185580, 1.255096909134953, 2.400721524717608, 3.376857807915165, 3.613693984917075, 4.664436544680546, 5.131350966148664, 5.904677737200461, 6.960076572555169, 7.464822166483185, 7.761510460664735, 8.558534996573224, 9.113987708376733, 9.507193260939047, 9.939277335287134, 10.78348061196340, 11.16176524211284, 11.97333604893348, 12.59563388731815, 13.15873457343860, 13.58889438949753, 14.20444896516345, 14.72319688510397, 15.25119960970857, 16.00289796795112
