L(s) = 1 | + (−11.3 + 0.338i)2-s + (127. − 7.64i)4-s + 93.0·5-s + 1.02e3i·7-s + (−1.44e3 + 129. i)8-s + (−1.05e3 + 31.4i)10-s − 1.54e3i·11-s + 5.67e3i·13-s + (−346. − 1.15e4i)14-s + (1.62e4 − 1.95e3i)16-s − 1.57e4i·17-s + 3.39e4·19-s + (1.18e4 − 712. i)20-s + (522. + 1.74e4i)22-s − 3.00e4·23-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0298i)2-s + (0.998 − 0.0597i)4-s + 0.333·5-s + 1.12i·7-s + (−0.995 + 0.0895i)8-s + (−0.332 + 0.00995i)10-s − 0.350i·11-s + 0.716i·13-s + (−0.0337 − 1.12i)14-s + (0.992 − 0.119i)16-s − 0.775i·17-s + 1.13·19-s + (0.332 − 0.0199i)20-s + (0.0104 + 0.349i)22-s − 0.515·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.314889 + 0.681544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.314889 + 0.681544i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (11.3 - 0.338i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 93.0T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.02e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 1.54e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 5.67e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.57e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.39e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.00e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.90e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.52e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.36e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.63e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 7.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.02e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.62e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.74e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 3.21e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 4.64e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.60e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 8.11e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.00e7iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 5.88e6T + 8.07e13T^{2} \) |
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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
−13.64152412446762530503372996819, −12.01677491623647111819068572449, −11.44699741149896189889999035081, −9.876915723411373538604617811593, −9.146369958418283503618541659880, −8.005075467942219722932604100301, −6.58854127475798998870758667890, −5.40994648702204758601477642326, −2.94948630607963530940716606353, −1.57429548016659894165046624903,
0.35289382712463266125796662765, 1.80749559453904822608551703544, 3.64664712749910141787997482602, 5.75180024537511799392602258421, 7.18252210082952676401713310904, 8.044058108460701331284184388254, 9.580760567487230370475354128843, 10.31066082969443865759061611562, 11.35588469027084449233617595560, 12.68911708133927301800611591659