L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.241 − 0.900i)3-s + (−0.499 + 0.866i)4-s + (−2.06 − 0.851i)5-s + (−0.658 + 0.658i)6-s + (−0.406 − 1.51i)7-s + 0.999·8-s + (1.84 − 1.06i)9-s + (0.296 + 2.21i)10-s − 2.93i·11-s + (0.900 + 0.241i)12-s + (−1.76 + 3.04i)13-s + (−1.11 + 1.11i)14-s + (−0.267 + 2.06i)15-s + (−0.5 − 0.866i)16-s + (−6.36 + 3.67i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.139 − 0.519i)3-s + (−0.249 + 0.433i)4-s + (−0.924 − 0.380i)5-s + (−0.269 + 0.269i)6-s + (−0.153 − 0.573i)7-s + 0.353·8-s + (0.615 − 0.355i)9-s + (0.0938 + 0.700i)10-s − 0.884i·11-s + (0.259 + 0.0696i)12-s + (−0.488 + 0.845i)13-s + (−0.297 + 0.297i)14-s + (−0.0690 + 0.533i)15-s + (−0.125 − 0.216i)16-s + (−1.54 + 0.890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0973086 + 0.393402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0973086 + 0.393402i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (2.06 + 0.851i)T \) |
| 37 | \( 1 + (2.51 + 5.53i)T \) |
good | 3 | \( 1 + (0.241 + 0.900i)T + (-2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (0.406 + 1.51i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + 2.93iT - 11T^{2} \) |
| 13 | \( 1 + (1.76 - 3.04i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (6.36 - 3.67i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.85 - 1.30i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + (-4.48 + 4.48i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3.51 - 3.51i)T + 31iT^{2} \) |
| 41 | \( 1 + (5.22 + 3.01i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 0.121T + 43T^{2} \) |
| 47 | \( 1 + (-6.92 + 6.92i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.14 - 4.27i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.71 + 13.8i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (12.9 - 3.46i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-9.85 + 2.64i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (3.60 - 6.23i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.65 - 4.65i)T - 73iT^{2} \) |
| 79 | \( 1 + (-2.20 + 0.591i)T + (68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-3.38 + 12.6i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (1.58 + 0.423i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 10.5iT - 97T^{2} \) |
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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
−10.91344241496102743103659992800, −10.14813937814146098243992084076, −8.854725101002082332796947827681, −8.256504295135300687701304006149, −7.13638777376424208685075035852, −6.33225869520695041644744005004, −4.37589917349505044034413530156, −3.85961009064519221859888269404, −1.96553056301890632077740146468, −0.30284025260101515416172942380,
2.51029848366842316997094735258, 4.30097156660221451183841742666, 4.89420123397317284334478852936, 6.42307720369898090575056430328, 7.24532315102894271286044616556, 8.139537992623255229279348691302, 9.117718857986778992713753695938, 10.14354838240977547411095185086, 10.73684408702900614893084209808, 11.86873936397296544600787299088