L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 3·11-s − 12-s + 13-s − 14-s + 16-s + 8·17-s + 18-s + 7·19-s + 21-s − 3·22-s + 5·23-s − 24-s − 5·25-s + 26-s − 27-s − 28-s + 4·29-s + 9·31-s + 32-s + 3·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 1.60·19-s + 0.218·21-s − 0.639·22-s + 1.04·23-s − 0.204·24-s − 25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s + 1.61·31-s + 0.176·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.353729214\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.353729214\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 41 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−15.63088525191993, −15.03228482789210, −14.32049432476221, −13.81160821798243, −13.36302561979157, −12.84590268539604, −12.13040363756687, −11.82870869708056, −11.40965196868548, −10.41452711065932, −10.19475885248610, −9.722275593091395, −8.827363013598007, −8.021197323490986, −7.476062962184801, −7.105756114747912, −6.038027956852107, −5.896792392799467, −5.071059422412314, −4.786373479153028, −3.703649965731662, −3.173742039038568, −2.632344694858133, −1.387573640059049, −0.7488403991667836,
0.7488403991667836, 1.387573640059049, 2.632344694858133, 3.173742039038568, 3.703649965731662, 4.786373479153028, 5.071059422412314, 5.896792392799467, 6.038027956852107, 7.105756114747912, 7.476062962184801, 8.021197323490986, 8.827363013598007, 9.722275593091395, 10.19475885248610, 10.41452711065932, 11.40965196868548, 11.82870869708056, 12.13040363756687, 12.84590268539604, 13.36302561979157, 13.81160821798243, 14.32049432476221, 15.03228482789210, 15.63088525191993