| L(s) = 1 | + (10.9 + 6.30i)2-s + (−7.65 − 13.2i)3-s + (15.5 + 26.8i)4-s + (−37.8 − 21.8i)5-s − 193. i·6-s + (534. + 733. i)7-s − 1.22e3i·8-s + (976. − 1.69e3i)9-s + (−275. − 477. i)10-s + (−2.78e3 + 1.60e3i)11-s + (237. − 411. i)12-s + (−7.92e3 − 3.49i)13-s + (1.20e3 + 1.13e4i)14-s + 668. i·15-s + (9.69e3 − 1.67e4i)16-s + (−1.13e4 − 1.96e4i)17-s + ⋯ |
| L(s) = 1 | + (0.965 + 0.557i)2-s + (−0.163 − 0.283i)3-s + (0.121 + 0.209i)4-s + (−0.135 − 0.0781i)5-s − 0.364i·6-s + (0.588 + 0.808i)7-s − 0.844i·8-s + (0.446 − 0.773i)9-s + (−0.0871 − 0.150i)10-s + (−0.629 + 0.363i)11-s + (0.0396 − 0.0686i)12-s + (−0.999 − 0.000441i)13-s + (0.117 + 1.10i)14-s + 0.0511i·15-s + (0.591 − 1.02i)16-s + (−0.559 − 0.969i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.70867 - 1.44223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70867 - 1.44223i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-534. - 733. i)T \) |
| 13 | \( 1 + (7.92e3 + 3.49i)T \) |
| good | 2 | \( 1 + (-10.9 - 6.30i)T + (64 + 110. i)T^{2} \) |
| 3 | \( 1 + (7.65 + 13.2i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (37.8 + 21.8i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (2.78e3 - 1.60e3i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.13e4 + 1.96e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-8.79e3 - 5.07e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.61e4 + 7.99e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.47e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.05e5 + 1.18e5i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (4.08e5 + 2.35e5i)T + (4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.80e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 1.65e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (9.34e4 + 5.39e4i)T + (2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-6.61e5 - 1.14e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.27e6 + 7.38e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (8.50e5 - 1.47e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (8.56e5 - 4.94e5i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 3.08e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (2.70e6 - 1.56e6i)T + (5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.25e6 + 2.16e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 4.71e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (-1.05e7 - 6.11e6i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 3.81e6iT - 8.07e13T^{2} \) |
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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
−12.41413087494520294730753834405, −11.97335681541593372735836644520, −10.23669826988264241494028844584, −9.071139928146277019018199760144, −7.51267205111876161194520732620, −6.50778858672614709983999032213, −5.24596956171445209707545314037, −4.40058140858407082207453844738, −2.51598465038287053865380368594, −0.51699395560306638106197184394,
1.74791105463988619833736346936, 3.30185353520662787294046894956, 4.57541726012563445349681240282, 5.28411534963695333194908112619, 7.29033632159809960595514791151, 8.294289187539818025628255885275, 10.11630183832941781729608573620, 10.97564090623903064636627212607, 11.80369495741927067470745273981, 13.18426687971623522655222091877
