L(s) = 1 | + 2·5-s − 3·11-s − 6·13-s + 4·17-s − 4·19-s + 4·23-s − 25-s − 5·29-s − 7·31-s − 2·41-s + 8·43-s + 2·47-s + 10·53-s − 6·55-s + 9·59-s + 8·61-s − 12·65-s − 6·67-s − 12·71-s + 11·73-s + 79-s + 15·83-s + 8·85-s + 10·89-s − 8·95-s + 5·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.904·11-s − 1.66·13-s + 0.970·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.928·29-s − 1.25·31-s − 0.312·41-s + 1.21·43-s + 0.291·47-s + 1.37·53-s − 0.809·55-s + 1.17·59-s + 1.02·61-s − 1.48·65-s − 0.733·67-s − 1.42·71-s + 1.28·73-s + 0.112·79-s + 1.64·83-s + 0.867·85-s + 1.05·89-s − 0.820·95-s + 0.507·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.790519020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.790519020\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−16.74294540497030, −16.12021172514580, −15.21562804119199, −14.78014526121447, −14.43661669916377, −13.62140064554781, −13.05702025815526, −12.66025836185616, −12.05299698830735, −11.30058332897854, −10.47481607244666, −10.23781050640288, −9.485626781972317, −9.082955717400622, −8.197831758886326, −7.403214624567161, −7.175538946968287, −6.124636787168636, −5.430992539920195, −5.165054012795119, −4.207506908961875, −3.277994198047089, −2.347957585507210, −2.003257879835304, −0.5948647549932412,
0.5948647549932412, 2.003257879835304, 2.347957585507210, 3.277994198047089, 4.207506908961875, 5.165054012795119, 5.430992539920195, 6.124636787168636, 7.175538946968287, 7.403214624567161, 8.197831758886326, 9.082955717400622, 9.485626781972317, 10.23781050640288, 10.47481607244666, 11.30058332897854, 12.05299698830735, 12.66025836185616, 13.05702025815526, 13.62140064554781, 14.43661669916377, 14.78014526121447, 15.21562804119199, 16.12021172514580, 16.74294540497030