| L(s) = 1 | − 2-s + 4-s − 3.23·5-s + 7-s − 8-s + 3.23·10-s − 11-s + 1.23·13-s − 14-s + 16-s + 6.47·17-s − 2.76·19-s − 3.23·20-s + 22-s − 4·23-s + 5.47·25-s − 1.23·26-s + 28-s + 4.47·29-s + 2·31-s − 32-s − 6.47·34-s − 3.23·35-s − 10.9·37-s + 2.76·38-s + 3.23·40-s − 6.47·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.44·5-s + 0.377·7-s − 0.353·8-s + 1.02·10-s − 0.301·11-s + 0.342·13-s − 0.267·14-s + 0.250·16-s + 1.56·17-s − 0.634·19-s − 0.723·20-s + 0.213·22-s − 0.834·23-s + 1.09·25-s − 0.242·26-s + 0.188·28-s + 0.830·29-s + 0.359·31-s − 0.176·32-s − 1.10·34-s − 0.546·35-s − 1.79·37-s + 0.448·38-s + 0.511·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.920696687811572017047939301580, −8.254430149250421489969947389261, −7.78285799199268548852351630789, −7.04147431222742331876374846261, −5.98266580793326831690460772674, −4.88349367678448981100309117625, −3.85633420373991417478843384162, −3.01716974034418228741824763660, −1.46213649792145244839479080657, 0,
1.46213649792145244839479080657, 3.01716974034418228741824763660, 3.85633420373991417478843384162, 4.88349367678448981100309117625, 5.98266580793326831690460772674, 7.04147431222742331876374846261, 7.78285799199268548852351630789, 8.254430149250421489969947389261, 8.920696687811572017047939301580
