L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 8-s + 9-s + 2·10-s − 2·11-s − 2·12-s − 4·13-s + 4·15-s + 16-s − 6·17-s − 18-s + 6·19-s − 2·20-s + 2·22-s + 8·23-s + 2·24-s − 25-s + 4·26-s + 4·27-s − 4·29-s − 4·30-s + 8·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.577·12-s − 1.10·13-s + 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s + 0.426·22-s + 1.66·23-s + 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.742·29-s − 0.730·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.980328152078102684123163989979, −7.33693276222809437900030588269, −6.77716700267734747408860579922, −5.95379406958284557994083924172, −4.98049558593349321269040276695, −4.63410545291689560595348798869, −3.25630907938759665003300195689, −2.42137707760054689964102686623, −0.889749452121683465172606934022, 0,
0.889749452121683465172606934022, 2.42137707760054689964102686623, 3.25630907938759665003300195689, 4.63410545291689560595348798869, 4.98049558593349321269040276695, 5.95379406958284557994083924172, 6.77716700267734747408860579922, 7.33693276222809437900030588269, 7.980328152078102684123163989979