| L(s) = 1 | + (−1.36 − 0.361i)2-s + (−1.45 − 0.938i)3-s + (1.73 + 0.989i)4-s + (−0.644 − 0.840i)5-s + (1.65 + 1.80i)6-s + (3.32 + 0.889i)7-s + (−2.01 − 1.98i)8-s + (1.23 + 2.73i)9-s + (0.577 + 1.38i)10-s + (−0.793 + 0.104i)11-s + (−1.60 − 3.07i)12-s + (0.0380 − 0.289i)13-s + (−4.21 − 2.41i)14-s + (0.149 + 1.82i)15-s + (2.04 + 3.43i)16-s + 7.22·17-s + ⋯ |
| L(s) = 1 | + (−0.966 − 0.255i)2-s + (−0.840 − 0.541i)3-s + (0.869 + 0.494i)4-s + (−0.288 − 0.375i)5-s + (0.673 + 0.738i)6-s + (1.25 + 0.336i)7-s + (−0.713 − 0.700i)8-s + (0.412 + 0.910i)9-s + (0.182 + 0.437i)10-s + (−0.239 + 0.0314i)11-s + (−0.462 − 0.886i)12-s + (0.0105 − 0.0801i)13-s + (−1.12 − 0.646i)14-s + (0.0387 + 0.472i)15-s + (0.510 + 0.859i)16-s + 1.75·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.549269 - 0.409991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.549269 - 0.409991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.361i)T \) |
| 3 | \( 1 + (1.45 + 0.938i)T \) |
| good | 5 | \( 1 + (0.644 + 0.840i)T + (-1.29 + 4.82i)T^{2} \) |
| 7 | \( 1 + (-3.32 - 0.889i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.793 - 0.104i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.0380 + 0.289i)T + (-12.5 - 3.36i)T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 + (2.42 + 5.85i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.46 + 5.46i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.49 - 1.15i)T + (7.50 + 28.0i)T^{2} \) |
| 31 | \( 1 + (5.11 + 2.95i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-10.4 - 4.33i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (3.64 - 0.976i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.0508 - 0.386i)T + (-41.5 + 11.1i)T^{2} \) |
| 47 | \( 1 + (-6.09 + 3.51i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.60 + 0.663i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (2.49 - 1.91i)T + (15.2 - 56.9i)T^{2} \) |
| 61 | \( 1 + (-4.52 + 5.90i)T + (-15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (1.18 - 9.02i)T + (-64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 11.2i)T + 71iT^{2} \) |
| 73 | \( 1 + (1.93 - 1.93i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.05 + 3.55i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 - 3.09i)T + (21.4 + 80.1i)T^{2} \) |
| 89 | \( 1 + (6.64 - 6.64i)T - 89iT^{2} \) |
| 97 | \( 1 + (5.82 + 10.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.51144384045829395230853867228, −10.85025682769210617835333402818, −9.889526654702053624451970386011, −8.452721458519249279815690634082, −7.982466313801591235955406842725, −6.95008426039007722879150654575, −5.70795395799339079274474519766, −4.55759362732491701765820726623, −2.38974949762008951851247479014, −0.909238142661243435342634224921,
1.40690956352760634729725291442, 3.65256832753137425452545662385, 5.21164220897010477596631839700, 6.01329634108249847102222871131, 7.46024777076512039486871212242, 7.935134780737494518823809078445, 9.333093772144943260767009789082, 10.25993415345808311103515773305, 10.90865187589801959059413236826, 11.61875109461493167962850514035
