| L(s) = 1 | + 3·3-s + 3·5-s + 2·7-s + 6·9-s + 9·15-s + 6·17-s + 4·19-s + 6·21-s + 23-s + 4·25-s + 9·27-s − 8·29-s − 7·31-s + 6·35-s + 37-s − 4·41-s + 6·43-s + 18·45-s − 8·47-s − 3·49-s + 18·51-s − 2·53-s + 12·57-s + 59-s + 4·61-s + 12·63-s + 5·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 0.755·7-s + 2·9-s + 2.32·15-s + 1.45·17-s + 0.917·19-s + 1.30·21-s + 0.208·23-s + 4/5·25-s + 1.73·27-s − 1.48·29-s − 1.25·31-s + 1.01·35-s + 0.164·37-s − 0.624·41-s + 0.914·43-s + 2.68·45-s − 1.16·47-s − 3/7·49-s + 2.52·51-s − 0.274·53-s + 1.58·57-s + 0.130·59-s + 0.512·61-s + 1.51·63-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.352875157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.352875157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.78588302111694005783178293791, −7.53469474940049042982286766442, −6.62698272420414288620906408737, −5.51527114674178017185987775950, −5.26037058011395678115848843425, −4.06496779216733125357263533537, −3.35126777930017754546321119820, −2.66805751036284509670553263148, −1.75927235992258079303127684934, −1.40602034173513172956519290617,
1.40602034173513172956519290617, 1.75927235992258079303127684934, 2.66805751036284509670553263148, 3.35126777930017754546321119820, 4.06496779216733125357263533537, 5.26037058011395678115848843425, 5.51527114674178017185987775950, 6.62698272420414288620906408737, 7.53469474940049042982286766442, 7.78588302111694005783178293791
