| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s − 13-s − 14-s + 16-s − 6·17-s − 18-s + 5·19-s + 21-s − 23-s − 24-s − 5·25-s + 26-s + 27-s + 28-s + 6·29-s − 31-s − 32-s + 6·34-s + 36-s + 11·37-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 + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.179·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.368052256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.368052256\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.78085571010523, −13.13845956669533, −12.61906824881724, −12.04864049673349, −11.53009724270232, −11.18071223661683, −10.66138451980956, −9.964952817072973, −9.675721046945309, −9.219583289175896, −8.563379052731439, −8.329105354682435, −7.636053242864174, −7.318199737622435, −6.771997009941941, −6.141029493826330, −5.587760311871155, −4.949031357366873, −4.116431829480785, −4.011344537156384, −2.867999006660508, −2.567357350003652, −1.983796133429409, −1.188367373122896, −0.5440895098492740,
0.5440895098492740, 1.188367373122896, 1.983796133429409, 2.567357350003652, 2.867999006660508, 4.011344537156384, 4.116431829480785, 4.949031357366873, 5.587760311871155, 6.141029493826330, 6.771997009941941, 7.318199737622435, 7.636053242864174, 8.329105354682435, 8.563379052731439, 9.219583289175896, 9.675721046945309, 9.964952817072973, 10.66138451980956, 11.18071223661683, 11.53009724270232, 12.04864049673349, 12.61906824881724, 13.13845956669533, 13.78085571010523
