| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s + 4·11-s − 12-s − 2·13-s − 15-s − 16-s − 18-s + 4·19-s + 20-s − 4·22-s + 3·24-s + 25-s + 2·26-s + 27-s + 2·29-s + 30-s − 5·32-s + 4·33-s − 36-s + 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.258·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.883·32-s + 0.696·33-s − 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4335 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4335 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358784537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358784537\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−8.324027666892170998148797227210, −7.88272580417467619012367098000, −7.16750910474070634391114159922, −6.44628802162953221998351414777, −5.26301391890678078722159871955, −4.47629935336188211298931611555, −3.84194557004788343759428710030, −2.95453421762719696432427292949, −1.69452749631301531601828409816, −0.75602502315476554931106499220,
0.75602502315476554931106499220, 1.69452749631301531601828409816, 2.95453421762719696432427292949, 3.84194557004788343759428710030, 4.47629935336188211298931611555, 5.26301391890678078722159871955, 6.44628802162953221998351414777, 7.16750910474070634391114159922, 7.88272580417467619012367098000, 8.324027666892170998148797227210
