| L(s) = 1 | + (−0.588 + 0.339i)2-s + (−63.7 + 110. i)4-s − 439. i·5-s + (−1.12e3 − 651. i)7-s − 173. i·8-s + (149. + 258. i)10-s + (−497. + 287. i)11-s + (−7.89e3 − 676. i)13-s + 884.·14-s + (−8.10e3 − 1.40e4i)16-s + (5.63e3 − 9.75e3i)17-s + (3.55e4 + 2.05e4i)19-s + (4.85e4 + 2.80e4i)20-s + (195. − 338. i)22-s + (2.21e4 + 3.83e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.0519 + 0.0300i)2-s + (−0.498 + 0.862i)4-s − 1.57i·5-s + (−1.24 − 0.717i)7-s − 0.119i·8-s + (0.0471 + 0.0816i)10-s + (−0.112 + 0.0651i)11-s + (−0.996 − 0.0854i)13-s + 0.0861·14-s + (−0.494 − 0.856i)16-s + (0.278 − 0.481i)17-s + (1.18 + 0.686i)19-s + (1.35 + 0.782i)20-s + (0.00390 − 0.00676i)22-s + (0.379 + 0.656i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0982 - 0.995i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0982 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5034649594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5034649594\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (7.89e3 + 676. i)T \) |
| good | 2 | \( 1 + (0.588 - 0.339i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 + 439. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (1.12e3 + 651. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (497. - 287. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-5.63e3 + 9.75e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-3.55e4 - 2.05e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.21e4 - 3.83e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (7.27e4 + 1.26e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.22e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-3.63e4 + 2.10e4i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-7.59e4 + 4.38e4i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (3.74e5 - 6.49e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 9.40e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 9.24e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (5.47e5 + 3.16e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-3.22e4 + 5.57e4i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.68e6 - 9.72e5i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-4.09e6 - 2.36e6i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.92e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 1.50e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.87e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (3.78e6 - 2.18e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.53e6 + 8.86e5i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.59946804088750102311511681367, −11.78082564408266295604325248306, −9.728553195241438962704455216669, −9.442337698677274268768931900799, −8.074971468807948852870891348229, −7.21112867334952972752449262045, −5.40449035158273329114657474350, −4.31629156105891714112261428518, −3.13898802353662946593203183013, −0.916197980471188526339690714067,
0.19529219998113051122989062891, 2.34564469928854032796999867939, 3.40349986971108909560659182754, 5.29134515730877017363337080100, 6.35978641404938618277012169093, 7.23053929185441033298920232950, 9.054240157059138436513810224028, 9.911003184392670824072272174341, 10.60726276200901698859947691212, 11.81616550793269916144738582027
