| L(s) = 1 | + 2-s + 4-s − 4·5-s − 4·7-s + 8-s − 4·10-s − 4·14-s + 16-s + 6·17-s + 19-s − 4·20-s − 6·23-s + 11·25-s − 4·28-s + 6·29-s + 10·31-s + 32-s + 6·34-s + 16·35-s − 12·37-s + 38-s − 4·40-s − 10·41-s − 4·43-s − 6·46-s + 6·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s − 1.51·7-s + 0.353·8-s − 1.26·10-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 0.894·20-s − 1.25·23-s + 11/5·25-s − 0.755·28-s + 1.11·29-s + 1.79·31-s + 0.176·32-s + 1.02·34-s + 2.70·35-s − 1.97·37-s + 0.162·38-s − 0.632·40-s − 1.56·41-s − 0.609·43-s − 0.884·46-s + 0.875·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.278565316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.278565316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.97556724975766, −14.18731322423308, −13.86247192237206, −13.24746677673062, −12.49957443156414, −12.24709843032891, −11.79652520548230, −11.59963446555315, −10.45901650788119, −10.18870072613009, −9.822776925557969, −8.658981902071964, −8.419673572663272, −7.740804818508451, −7.168686118547219, −6.701414849227935, −6.189193454190015, −5.393343827472900, −4.801983876170647, −4.018566226700316, −3.613630437081059, −3.169341818063843, −2.623763680352895, −1.296019856842938, −0.3918107643370827,
0.3918107643370827, 1.296019856842938, 2.623763680352895, 3.169341818063843, 3.613630437081059, 4.018566226700316, 4.801983876170647, 5.393343827472900, 6.189193454190015, 6.701414849227935, 7.168686118547219, 7.740804818508451, 8.419673572663272, 8.658981902071964, 9.822776925557969, 10.18870072613009, 10.45901650788119, 11.59963446555315, 11.79652520548230, 12.24709843032891, 12.49957443156414, 13.24746677673062, 13.86247192237206, 14.18731322423308, 14.97556724975766
