L(s) = 1 | + 3-s − 5-s + 9-s + 11-s + 7·13-s − 15-s + 2·17-s − 2·23-s + 25-s + 27-s + 2·29-s + 3·31-s + 33-s + 12·37-s + 7·39-s + 6·41-s − 43-s − 45-s + 10·47-s + 2·51-s − 55-s − 7·59-s − 7·65-s + 4·67-s − 2·69-s − 9·71-s + 9·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.94·13-s − 0.258·15-s + 0.485·17-s − 0.417·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.538·31-s + 0.174·33-s + 1.97·37-s + 1.12·39-s + 0.937·41-s − 0.152·43-s − 0.149·45-s + 1.45·47-s + 0.280·51-s − 0.134·55-s − 0.911·59-s − 0.868·65-s + 0.488·67-s − 0.240·69-s − 1.06·71-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.577701119\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.577701119\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.52475319668425, −13.08429638607244, −12.57891490280706, −12.04610896318937, −11.57273318558088, −11.00422401842054, −10.72708519616233, −10.08441084312749, −9.476880296561730, −9.101975922533578, −8.526397400110023, −8.114041761782120, −7.756653824334342, −7.129962627514689, −6.490606726254446, −5.995673882679352, −5.671739756149029, −4.674544143665039, −4.209263851915511, −3.813188215481167, −3.200727698620536, −2.689710581249765, −1.903759050294294, −1.108421836407083, −0.7356496715920607,
0.7356496715920607, 1.108421836407083, 1.903759050294294, 2.689710581249765, 3.200727698620536, 3.813188215481167, 4.209263851915511, 4.674544143665039, 5.671739756149029, 5.995673882679352, 6.490606726254446, 7.129962627514689, 7.756653824334342, 8.114041761782120, 8.526397400110023, 9.101975922533578, 9.476880296561730, 10.08441084312749, 10.72708519616233, 11.00422401842054, 11.57273318558088, 12.04610896318937, 12.57891490280706, 13.08429638607244, 13.52475319668425