L(s) = 1 | − 3-s − 2·7-s + 9-s − 2·11-s − 13-s + 2·17-s − 2·19-s + 2·21-s + 8·23-s − 27-s + 6·29-s − 2·31-s + 2·33-s + 2·37-s + 39-s − 2·41-s − 6·47-s − 3·49-s − 2·51-s − 10·53-s + 2·57-s + 14·59-s + 10·61-s − 2·63-s + 2·67-s − 8·69-s − 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s − 0.312·41-s − 0.875·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s + 0.264·57-s + 1.82·59-s + 1.28·61-s − 0.251·63-s + 0.244·67-s − 0.963·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.43508229365475, −14.16974688527815, −13.08908434908523, −13.03235519813182, −12.74127295616538, −11.91826508219749, −11.53239694979891, −10.95795842697769, −10.45773408565540, −9.903801114072905, −9.640374114811199, −8.824524823837537, −8.397348371128619, −7.709956354415738, −7.109567568789629, −6.650663330490241, −6.210956046607339, −5.421210417094740, −5.074403703604780, −4.471651618477418, −3.702057198543876, −3.019242498830125, −2.591797166228264, −1.593759194491513, −0.8043424375354338, 0,
0.8043424375354338, 1.593759194491513, 2.591797166228264, 3.019242498830125, 3.702057198543876, 4.471651618477418, 5.074403703604780, 5.421210417094740, 6.210956046607339, 6.650663330490241, 7.109567568789629, 7.709956354415738, 8.397348371128619, 8.824524823837537, 9.640374114811199, 9.903801114072905, 10.45773408565540, 10.95795842697769, 11.53239694979891, 11.91826508219749, 12.74127295616538, 13.03235519813182, 13.08908434908523, 14.16974688527815, 14.43508229365475