| L(s) = 1 | + (135. − 234. i)3-s + (−684. − 1.18e3i)5-s + (−6.32e3 + 558. i)7-s + (−2.68e4 − 4.64e4i)9-s + (−5.46e3 + 9.46e3i)11-s − 7.96e3·13-s − 3.70e5·15-s + (1.62e5 − 2.80e5i)17-s + (−1.15e5 − 2.00e5i)19-s + (−7.25e5 + 1.55e6i)21-s + (1.05e6 + 1.82e6i)23-s + (3.87e4 − 6.70e4i)25-s − 9.20e6·27-s − 5.46e6·29-s + (−9.89e5 + 1.71e6i)31-s + ⋯ |
| L(s) = 1 | + (0.965 − 1.67i)3-s + (−0.489 − 0.848i)5-s + (−0.996 + 0.0879i)7-s + (−1.36 − 2.36i)9-s + (−0.112 + 0.194i)11-s − 0.0773·13-s − 1.89·15-s + (0.470 − 0.814i)17-s + (−0.204 − 0.353i)19-s + (−0.814 + 1.75i)21-s + (0.784 + 1.35i)23-s + (0.0198 − 0.0343i)25-s − 3.33·27-s − 1.43·29-s + (−0.192 + 0.333i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0247 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8810211211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8810211211\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (6.32e3 - 558. i)T \) |
| good | 3 | \( 1 + (-135. + 234. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (684. + 1.18e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (5.46e3 - 9.46e3i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 7.96e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.62e5 + 2.80e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.15e5 + 2.00e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.05e6 - 1.82e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 5.46e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (9.89e5 - 1.71e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (2.38e6 + 4.12e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 5.57e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.15e6 - 3.73e6i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.51e7 + 7.81e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (3.64e7 - 6.31e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-3.71e6 - 6.43e6i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.28e7 + 9.16e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 1.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (1.02e8 - 1.77e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-4.43e7 - 7.68e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 2.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.74e8 + 4.75e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.16e9T + 7.60e17T^{2} \) |
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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
−11.54698744489577951585338309308, −9.447562581363786192803180129031, −8.832395083549511416752636178350, −7.62106395960875406262818115854, −7.00694742079802958705365364420, −5.62089406241809985595787483084, −3.63258770019610523677960048234, −2.54218245694974252286187518227, −1.17575387596188296463319455770, −0.20449819085918260279422807843,
2.60317494982930376849903997559, 3.43001663674976718581226579243, 4.22665674854952222474818469175, 5.79774355525872455532178322918, 7.39614854572715643124138604191, 8.598716405384883126047253050615, 9.532352742322311279316823223218, 10.48046915864425606637675070191, 11.01241355489978421212809960713, 12.73941523902980902703123226507
