| L(s) = 1 | + (−0.779 + 1.54i)3-s + (0.268 − 2.21i)5-s − 2.49·7-s + (−1.78 − 2.41i)9-s + 3.17·11-s − 5.31i·13-s + (3.22 + 2.14i)15-s + 2.70i·17-s + 8.13i·19-s + (1.94 − 3.86i)21-s + (−4.12 + 2.44i)23-s + (−4.85 − 1.19i)25-s + (5.12 − 0.877i)27-s + 2.82i·29-s − 6.28·31-s + ⋯ |
| L(s) = 1 | + (−0.450 + 0.892i)3-s + (0.119 − 0.992i)5-s − 0.944·7-s + (−0.594 − 0.804i)9-s + 0.958·11-s − 1.47i·13-s + (0.832 + 0.554i)15-s + 0.655i·17-s + 1.86i·19-s + (0.425 − 0.843i)21-s + (−0.859 + 0.510i)23-s + (−0.971 − 0.238i)25-s + (0.985 − 0.168i)27-s + 0.524i·29-s − 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1338527355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1338527355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.779 - 1.54i)T \) |
| 5 | \( 1 + (-0.268 + 2.21i)T \) |
| 23 | \( 1 + (4.12 - 2.44i)T \) |
| good | 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 5.31iT - 13T^{2} \) |
| 17 | \( 1 - 2.70iT - 17T^{2} \) |
| 19 | \( 1 - 8.13iT - 19T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 + 3.17iT - 41T^{2} \) |
| 43 | \( 1 + 6.93T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 - 3.79iT - 53T^{2} \) |
| 59 | \( 1 - 11.2iT - 59T^{2} \) |
| 61 | \( 1 + 0.443iT - 61T^{2} \) |
| 67 | \( 1 - 6.68T + 67T^{2} \) |
| 71 | \( 1 + 14.4iT - 71T^{2} \) |
| 73 | \( 1 - 2.80iT - 73T^{2} \) |
| 79 | \( 1 + 6.08iT - 79T^{2} \) |
| 83 | \( 1 + 8.29iT - 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−10.08382881813111559716750535262, −9.270695779396516446977648682509, −8.569448766122983200512045362431, −7.72563994458744655994015159563, −6.29424602751306239756847965533, −5.82211024056760612837385940936, −5.05893922672439677936550198090, −3.77693283773733980969363634579, −3.49641285832345769133320778818, −1.50662548059736547743559551698,
0.05672534687185525080570051908, 1.84397593016784702138097402860, 2.78099353777430101827741129351, 3.89231921100894508466692001359, 5.10179507175226642592033053306, 6.35525771080209051862009127259, 6.73438316808660250594767743203, 7.06809336886233007719917799294, 8.323244109965764916550816912717, 9.356710477388307481847978755312
