| L(s) = 1 | + (−10.6 + 3.86i)2-s + (98.0 − 82.2i)4-s + 192. i·5-s + 343i·7-s + (−724. + 1.25e3i)8-s + (−746. − 2.04e3i)10-s − 6.37e3·11-s − 4.80e3·13-s + (−1.32e3 − 3.64e3i)14-s + (2.84e3 − 1.61e4i)16-s − 178. i·17-s − 2.14e4i·19-s + (1.58e4 + 1.89e4i)20-s + (6.78e4 − 2.46e4i)22-s − 4.58e4·23-s + ⋯ |
| L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + 0.689i·5-s + 0.377i·7-s + (−0.499 + 0.866i)8-s + (−0.235 − 0.648i)10-s − 1.44·11-s − 0.607·13-s + (−0.129 − 0.355i)14-s + (0.173 − 0.984i)16-s − 0.00880i·17-s − 0.716i·19-s + (0.443 + 0.528i)20-s + (1.35 − 0.494i)22-s − 0.786·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0825i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7517442367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7517442367\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.6 - 3.86i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 343iT \) |
| good | 5 | \( 1 - 192. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + 6.37e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.80e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 178. iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.14e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 4.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.22e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 4.17e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 9.08e3T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.88e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 4.52e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.81e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.26e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.49e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.34e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.65e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.77e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.78e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 3.59e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.52e6iT - 4.42e13T^{2} \) |
| 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
−10.57762577365540420723168320970, −9.930033779969738025991707324319, −8.828508972613601056849622636511, −7.82924836802832765927679146738, −7.08297011328491698729009598441, −5.97537995656589536290045144983, −4.95465071229474763678499532630, −2.92232505980260544167566390944, −2.14411498819095511310133922594, −0.38940317097698748693711674417,
0.60847769383905359963494350672, 1.88640820430866652643054645730, 3.07670330009188717955048776857, 4.52545487672385910922278224561, 5.79449146965897934722419602335, 7.22922632609246312226491088927, 7.984441873681398270524430198663, 8.821010845889541679579007579874, 9.988260519167315966466393790258, 10.47896288989939188817643296388
