L(s) = 1 | + 1.48·2-s + 1.17·3-s + 0.211·4-s + 5-s + 1.75·6-s + 4.14·7-s − 2.65·8-s − 1.61·9-s + 1.48·10-s − 6.52·11-s + 0.249·12-s − 3.98·13-s + 6.16·14-s + 1.17·15-s − 4.37·16-s + 17-s − 2.39·18-s + 3.41·19-s + 0.211·20-s + 4.88·21-s − 9.70·22-s − 2.43·23-s − 3.13·24-s + 25-s − 5.92·26-s − 5.43·27-s + 0.877·28-s + ⋯ |
L(s) = 1 | + 1.05·2-s + 0.680·3-s + 0.105·4-s + 0.447·5-s + 0.715·6-s + 1.56·7-s − 0.940·8-s − 0.537·9-s + 0.470·10-s − 1.96·11-s + 0.0719·12-s − 1.10·13-s + 1.64·14-s + 0.304·15-s − 1.09·16-s + 0.242·17-s − 0.565·18-s + 0.782·19-s + 0.0473·20-s + 1.06·21-s − 2.07·22-s − 0.507·23-s − 0.639·24-s + 0.200·25-s − 1.16·26-s − 1.04·27-s + 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 3 | \( 1 - 1.17T + 3T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 + 6.52T + 11T^{2} \) |
| 13 | \( 1 + 3.98T + 13T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 - 1.91T + 29T^{2} \) |
| 31 | \( 1 + 4.32T + 31T^{2} \) |
| 37 | \( 1 - 2.88T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 + 0.623T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 3.13T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 9.73T + 67T^{2} \) |
| 73 | \( 1 + 9.97T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 9.58T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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
−7.72341332847895532905229529741, −7.21402463122411724020711964245, −5.76326064056654234750650209550, −5.43429020596256742188118863566, −4.92824527356648780085138531733, −4.23192242404317363471322576864, −3.02212611963316891071802271963, −2.64866826537075028252854861650, −1.79420394899634640565497618428, 0,
1.79420394899634640565497618428, 2.64866826537075028252854861650, 3.02212611963316891071802271963, 4.23192242404317363471322576864, 4.92824527356648780085138531733, 5.43429020596256742188118863566, 5.76326064056654234750650209550, 7.21402463122411724020711964245, 7.72341332847895532905229529741