| L(s) = 1 | + 7-s − 6·11-s − 5·13-s − 3·17-s + 3·23-s − 5·25-s + 9·29-s − 4·31-s − 2·37-s − 8·43-s − 6·49-s − 3·53-s − 9·59-s − 10·61-s + 5·67-s + 6·71-s − 7·73-s − 6·77-s − 10·79-s − 6·83-s − 12·89-s − 5·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.80·11-s − 1.38·13-s − 0.727·17-s + 0.625·23-s − 25-s + 1.67·29-s − 0.718·31-s − 0.328·37-s − 1.21·43-s − 6/7·49-s − 0.412·53-s − 1.17·59-s − 1.28·61-s + 0.610·67-s + 0.712·71-s − 0.819·73-s − 0.683·77-s − 1.12·79-s − 0.658·83-s − 1.27·89-s − 0.524·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3185838757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3185838757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.49652715750582, −13.90729496984546, −13.48437467399350, −12.91858628755081, −12.46971312640225, −12.01669725743208, −11.27661934658849, −10.94293102408641, −10.24796793426786, −9.957103497647763, −9.384329262447714, −8.547194477037981, −8.249442636527130, −7.536053612998843, −7.285128677200876, −6.532874841534245, −5.866792343326652, −5.113428294798039, −4.870382142114341, −4.354856108305304, −3.278591421878439, −2.769538282563847, −2.213691438956763, −1.483592937657245, −0.1862705959458297,
0.1862705959458297, 1.483592937657245, 2.213691438956763, 2.769538282563847, 3.278591421878439, 4.354856108305304, 4.870382142114341, 5.113428294798039, 5.866792343326652, 6.532874841534245, 7.285128677200876, 7.536053612998843, 8.249442636527130, 8.547194477037981, 9.384329262447714, 9.957103497647763, 10.24796793426786, 10.94293102408641, 11.27661934658849, 12.01669725743208, 12.46971312640225, 12.91858628755081, 13.48437467399350, 13.90729496984546, 14.49652715750582
