L(s) = 1 | + 2·5-s − 6·11-s − 6·13-s + 2·17-s + 4·19-s + 2·23-s − 25-s − 8·29-s − 4·31-s + 6·37-s − 10·41-s + 4·43-s + 4·47-s + 4·53-s − 12·55-s − 12·59-s − 2·61-s − 12·65-s − 12·67-s + 6·71-s + 2·73-s − 8·79-s + 4·85-s + 14·89-s + 8·95-s + 2·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.80·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.417·23-s − 1/5·25-s − 1.48·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 0.583·47-s + 0.549·53-s − 1.61·55-s − 1.56·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s + 0.712·71-s + 0.234·73-s − 0.900·79-s + 0.433·85-s + 1.48·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.340276107\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.340276107\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.07501456997743, −14.80732197654590, −14.08895761384143, −13.49985299940254, −13.23520796262457, −12.53717554325225, −12.19528733824409, −11.40424451908074, −10.82633881006276, −10.21188802430051, −9.832917432211684, −9.415755257713374, −8.758849395598375, −7.821776343201235, −7.476634067463453, −7.175221952350288, −5.988856869116287, −5.676896061232039, −5.046708964115618, −4.711064792332173, −3.540953280957518, −2.866722517258167, −2.304253481051797, −1.678260613403831, −0.4217586548308436,
0.4217586548308436, 1.678260613403831, 2.304253481051797, 2.866722517258167, 3.540953280957518, 4.711064792332173, 5.046708964115618, 5.676896061232039, 5.988856869116287, 7.175221952350288, 7.476634067463453, 7.821776343201235, 8.758849395598375, 9.415755257713374, 9.832917432211684, 10.21188802430051, 10.82633881006276, 11.40424451908074, 12.19528733824409, 12.53717554325225, 13.23520796262457, 13.49985299940254, 14.08895761384143, 14.80732197654590, 15.07501456997743