| L(s) = 1 | + 7.77i·2-s − 62.8·3-s + 67.5·4-s − 184. i·5-s − 488. i·6-s + 1.07e3·7-s + 1.52e3i·8-s + 1.76e3·9-s + 1.43e3·10-s − 3.54e3·11-s − 4.24e3·12-s + 6.31e3i·13-s + 8.32e3i·14-s + 1.15e4i·15-s − 3.16e3·16-s + 2.23e4i·17-s + ⋯ |
| L(s) = 1 | + 0.687i·2-s − 1.34·3-s + 0.527·4-s − 0.658i·5-s − 0.923i·6-s + 1.18·7-s + 1.04i·8-s + 0.806·9-s + 0.452·10-s − 0.802·11-s − 0.709·12-s + 0.797i·13-s + 0.811i·14-s + 0.885i·15-s − 0.193·16-s + 1.10i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.719283 + 1.00248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.719283 + 1.00248i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (9.86e4 + 2.91e5i)T \) |
| good | 2 | \( 1 - 7.77iT - 128T^{2} \) |
| 3 | \( 1 + 62.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 184. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 1.07e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.31e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.23e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 9.60e3iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 9.59e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 2.48e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.51e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 - 4.49e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.85e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 2.02e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.33e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.79e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.60e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.14e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.02e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.45e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 3.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.64e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 8.57e6iT - 8.07e13T^{2} \) |
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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
−15.56918895623137453472058311193, −14.26519419510154689057804065807, −12.53444218669063078394868558287, −11.44718378573069610129706035254, −10.73403126298811753256613983377, −8.525823998841166835627233557065, −7.19597361861257997117537328488, −5.71908040028696602537521505402, −4.89277729687217378098491751682, −1.53944211701094299517509770700,
0.67235719119190520254172320125, 2.59505982484574304184004601267, 4.90613822179769583072035439577, 6.34142764209240287408687751283, 7.74868463314766826715530600543, 10.24942850117848740921886308702, 10.98100073526202889279725152643, 11.66117534667398191843754672135, 12.79678343283901823821188072831, 14.59476285821922312812103077969
