| L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s − 3·8-s + 9-s − 10-s + 12-s + 6·13-s + 15-s − 16-s − 2·17-s + 18-s + 8·19-s + 20-s + 8·23-s + 3·24-s + 25-s + 6·26-s − 27-s − 2·29-s + 30-s − 4·31-s + 5·32-s − 2·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 1.66·23-s + 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.718·31-s + 0.883·32-s − 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437375972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437375972\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−10.66117211895074937453677481475, −9.357583073556529155557794013065, −8.843078054411908501657519136744, −7.69164818258356184893378968692, −6.67009631001529545506185276841, −5.69208011630684437322158261218, −5.01564976002989168468776414543, −3.94068846921577588199342611811, −3.18167275817234212778601509206, −0.982469269817440515290174005672,
0.982469269817440515290174005672, 3.18167275817234212778601509206, 3.94068846921577588199342611811, 5.01564976002989168468776414543, 5.69208011630684437322158261218, 6.67009631001529545506185276841, 7.69164818258356184893378968692, 8.843078054411908501657519136744, 9.357583073556529155557794013065, 10.66117211895074937453677481475
