| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s + 9-s − 2·10-s + 6·11-s − 2·12-s + 3·13-s − 15-s − 4·16-s − 4·17-s − 2·18-s + 19-s + 2·20-s − 12·22-s + 25-s − 6·26-s − 27-s − 8·29-s + 2·30-s − 31-s + 8·32-s − 6·33-s + 8·34-s + 2·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 1/3·9-s − 0.632·10-s + 1.80·11-s − 0.577·12-s + 0.832·13-s − 0.258·15-s − 16-s − 0.970·17-s − 0.471·18-s + 0.229·19-s + 0.447·20-s − 2.55·22-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 1.48·29-s + 0.365·30-s − 0.179·31-s + 1.41·32-s − 1.04·33-s + 1.37·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8122356483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8122356483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.29908175666181, −11.79225261469477, −11.32142403526158, −11.03716682735918, −10.72624965702733, −10.12170163641057, −9.547245903179929, −9.371145773463532, −8.836329402208436, −8.673739368799404, −7.969197244676666, −7.392003331767203, −6.986417397921052, −6.557442402019590, −6.180028697049588, −5.693752058898321, −5.039816775798925, −4.396895893542904, −3.977616631404176, −3.482124906371101, −2.639374393102073, −1.772386479259701, −1.639867765266102, −1.069831393753649, −0.3382050895789489,
0.3382050895789489, 1.069831393753649, 1.639867765266102, 1.772386479259701, 2.639374393102073, 3.482124906371101, 3.977616631404176, 4.396895893542904, 5.039816775798925, 5.693752058898321, 6.180028697049588, 6.557442402019590, 6.986417397921052, 7.392003331767203, 7.969197244676666, 8.673739368799404, 8.836329402208436, 9.371145773463532, 9.547245903179929, 10.12170163641057, 10.72624965702733, 11.03716682735918, 11.32142403526158, 11.79225261469477, 12.29908175666181
