| L(s) = 1 | + (1 − 2i)5-s − i·7-s − 2·11-s − 2i·13-s − 8i·17-s − 2·19-s + (−3 − 4i)25-s − 6·29-s − 6·31-s + (−2 − i)35-s + 8i·37-s − 6·41-s + 8i·43-s + 4i·47-s − 49-s + ⋯ |
| L(s) = 1 | + (0.447 − 0.894i)5-s − 0.377i·7-s − 0.603·11-s − 0.554i·13-s − 1.94i·17-s − 0.458·19-s + (−0.600 − 0.800i)25-s − 1.11·29-s − 1.07·31-s + (−0.338 − 0.169i)35-s + 1.31i·37-s − 0.937·41-s + 1.21i·43-s + 0.583i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5395626397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5395626397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−7.78637554285595287807837674701, −7.23982752453284554159702095117, −6.34939921314195853668822334234, −5.38835197531502580602532128080, −5.06548870694745889855017109045, −4.21799697076741159139333303547, −3.16732031872953320590923270595, −2.31092564693398283615615400929, −1.19493067099303979779355492198, −0.13876351859999194898018041976,
1.88867449626509289012333985965, 2.17642524197737091710857923314, 3.50282581088394799723528276198, 3.93061120079318067017096599878, 5.22499556027697807107143909652, 5.79806467560775040801618813002, 6.44543625275410000622458986563, 7.18266884366156334835730297336, 7.86202412993091424911309139482, 8.721615533959288411450524021992
