| L(s) = 1 | + (0.319 + 11.3i)2-s + (−127. + 7.23i)4-s − 412.·5-s − 359. i·7-s + (−122. − 1.44e3i)8-s + (−131. − 4.66e3i)10-s − 3.81e3i·11-s + 1.41e4i·13-s + (4.06e3 − 114. i)14-s + (1.62e4 − 1.84e3i)16-s − 3.33e3i·17-s + 3.43e4·19-s + (5.26e4 − 2.98e3i)20-s + (4.31e4 − 1.22e3i)22-s − 6.99e4·23-s + ⋯ |
| L(s) = 1 | + (0.0282 + 0.999i)2-s + (−0.998 + 0.0565i)4-s − 1.47·5-s − 0.395i·7-s + (−0.0847 − 0.996i)8-s + (−0.0416 − 1.47i)10-s − 0.865i·11-s + 1.78i·13-s + (0.395 − 0.0111i)14-s + (0.993 − 0.112i)16-s − 0.164i·17-s + 1.14·19-s + (1.47 − 0.0833i)20-s + (0.864 − 0.0244i)22-s − 1.19·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.984774 + 0.359630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.984774 + 0.359630i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.319 - 11.3i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 412.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 359. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 3.81e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.41e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.33e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.43e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.99e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.86e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.89e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 1.90e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 1.59e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 7.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.35e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.32e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.98e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.37e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.28e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.12e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.21e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.85e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 4.75e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 7.08e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 1.34e7T + 8.07e13T^{2} \) |
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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
−13.75617628751371223720862684763, −12.15051264832201724292173721350, −11.31570908266393886202317788905, −9.594174057033727687258455496758, −8.351036864985588503955274716730, −7.49513098054304098454329204367, −6.35739365044282014892813705137, −4.58303014196294269866863378142, −3.65478838437827727676905715779, −0.59771067747753847280758384034,
0.78531100783158862261039860231, 2.81464974484595904364071948565, 4.00507743172410440628580090862, 5.33416254880103199549607734908, 7.59327322445352699958484476493, 8.439560598589464462998426506043, 9.937396600557531529008529878053, 10.93511233993161292935499664554, 12.25111259439356022222822671521, 12.39820370453609338558022432893
