| L(s) = 1 | − 1.07e3i·5-s + (−436. + 2.36e3i)7-s + 1.05e4·11-s − 2.46e4i·13-s − 6.36e4i·17-s + 1.36e5i·19-s + 3.40e5·23-s − 7.57e5·25-s + 9.67e5·29-s − 9.44e4i·31-s + (2.53e6 + 4.68e5i)35-s + 7.33e5·37-s − 2.00e6i·41-s + 5.98e6·43-s − 1.51e6i·47-s + ⋯ |
| L(s) = 1 | − 1.71i·5-s + (−0.182 + 0.983i)7-s + 0.722·11-s − 0.863i·13-s − 0.762i·17-s + 1.04i·19-s + 1.21·23-s − 1.93·25-s + 1.36·29-s − 0.102i·31-s + (1.68 + 0.312i)35-s + 0.391·37-s − 0.710i·41-s + 1.74·43-s − 0.310i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.100434578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.100434578\) |
| \(L(5)\) |
|
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 | \( 1 + (436. - 2.36e3i)T \) |
| good | 5 | \( 1 + 1.07e3iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 1.05e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 2.46e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 6.36e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.36e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 3.40e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 9.67e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 9.44e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 7.33e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.00e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.98e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 1.51e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 3.99e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 9.18e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.02e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 4.55e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.39e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 3.93e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 7.44e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 1.51e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.77e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 8.92e7iT - 7.83e15T^{2} \) |
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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
−10.14594620905684241598386260058, −9.072591152843767128184811463055, −8.721227842784236272946580937033, −7.60536957360887929486472618867, −6.04387232829523298064655637712, −5.27884422115012646694623685280, −4.33392202923559186796586497414, −2.87552714135133931439123944368, −1.42319493034848362426543821497, −0.53256416305235541249355354379,
1.06411278524031688825679864643, 2.54016045849768746117732967692, 3.52419774869759986091098373246, 4.52044827201450632357228105339, 6.41317256872989140946466558970, 6.74712634334704130784131855778, 7.66953657810014781029724580377, 9.114982020148608366087511345599, 10.09542915698029837878360887554, 10.96123565864501981954270684681
