| L(s) = 1 | − 3-s − 4·5-s − 2·7-s + 9-s − 11-s + 4·15-s − 17-s + 2·21-s − 6·23-s + 11·25-s − 27-s + 2·29-s + 4·31-s + 33-s + 8·35-s − 2·37-s − 6·41-s + 4·43-s − 4·45-s − 6·47-s − 3·49-s + 51-s − 8·53-s + 4·55-s − 8·59-s + 8·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.03·15-s − 0.242·17-s + 0.436·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.174·33-s + 1.35·35-s − 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.596·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s − 1.09·53-s + 0.539·55-s − 1.04·59-s + 1.02·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.32667781511200, −14.89003643226253, −14.08452435730610, −13.62251516097880, −12.78722817297280, −12.48380179491902, −12.08202762962475, −11.37145544804685, −11.25202982919743, −10.41788764242255, −10.01189528884366, −9.366466088427790, −8.550102144839762, −8.099105586599944, −7.695306345386725, −6.915887418546358, −6.590578287461721, −5.916682330147906, −5.088886247149911, −4.517510308949704, −4.000405268449657, −3.362111918901243, −2.827770184735107, −1.720784981238943, −0.6092641087973046, 0,
0.6092641087973046, 1.720784981238943, 2.827770184735107, 3.362111918901243, 4.000405268449657, 4.517510308949704, 5.088886247149911, 5.916682330147906, 6.590578287461721, 6.915887418546358, 7.695306345386725, 8.099105586599944, 8.550102144839762, 9.366466088427790, 10.01189528884366, 10.41788764242255, 11.25202982919743, 11.37145544804685, 12.08202762962475, 12.48380179491902, 12.78722817297280, 13.62251516097880, 14.08452435730610, 14.89003643226253, 15.32667781511200
