L(s) = 1 | + 2-s + 4-s + 4·5-s + 3·7-s + 8-s + 4·10-s − 2·11-s + 13-s + 3·14-s + 16-s − 3·17-s + 4·20-s − 2·22-s + 23-s + 11·25-s + 26-s + 3·28-s − 5·29-s + 8·31-s + 32-s − 3·34-s + 12·35-s + 2·37-s + 4·40-s − 8·41-s + 4·43-s − 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 1.13·7-s + 0.353·8-s + 1.26·10-s − 0.603·11-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s + 0.894·20-s − 0.426·22-s + 0.208·23-s + 11/5·25-s + 0.196·26-s + 0.566·28-s − 0.928·29-s + 1.43·31-s + 0.176·32-s − 0.514·34-s + 2.02·35-s + 0.328·37-s + 0.632·40-s − 1.24·41-s + 0.609·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.425586746\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.425586746\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.066304209751900883680372833625, −7.08630692294978510791408292834, −6.40001715187249414844342874557, −5.83278378320473922139761712408, −4.97804072056498150514618992266, −4.85908639596030014967981752307, −3.61504607409621719549683932368, −2.49657072322899130135959102215, −2.06231205992969271621001293694, −1.18389774049284853047939869594,
1.18389774049284853047939869594, 2.06231205992969271621001293694, 2.49657072322899130135959102215, 3.61504607409621719549683932368, 4.85908639596030014967981752307, 4.97804072056498150514618992266, 5.83278378320473922139761712408, 6.40001715187249414844342874557, 7.08630692294978510791408292834, 8.066304209751900883680372833625