| L(s) = 1 | + (−1.41 − 0.0594i)2-s + (0.991 − 0.130i)3-s + (1.99 + 0.167i)4-s + (3.05 + 0.402i)5-s + (−1.40 + 0.125i)6-s + (−1.13 + 2.38i)7-s + (−2.80 − 0.355i)8-s + (0.965 − 0.258i)9-s + (−4.29 − 0.749i)10-s + (1.78 + 1.36i)11-s + (1.99 − 0.0936i)12-s + (0.152 − 0.367i)13-s + (1.74 − 3.30i)14-s + 3.08·15-s + (3.94 + 0.669i)16-s + (1.92 + 3.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0420i)2-s + (0.572 − 0.0753i)3-s + (0.996 + 0.0839i)4-s + (1.36 + 0.179i)5-s + (−0.575 + 0.0512i)6-s + (−0.429 + 0.903i)7-s + (−0.992 − 0.125i)8-s + (0.321 − 0.0862i)9-s + (−1.35 − 0.237i)10-s + (0.537 + 0.412i)11-s + (0.576 − 0.0270i)12-s + (0.0422 − 0.101i)13-s + (0.466 − 0.884i)14-s + 0.795·15-s + (0.985 + 0.167i)16-s + (0.466 + 0.807i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35936 + 0.515216i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35936 + 0.515216i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0594i)T \) |
| 3 | \( 1 + (-0.991 + 0.130i)T \) |
| 7 | \( 1 + (1.13 - 2.38i)T \) |
| good | 5 | \( 1 + (-3.05 - 0.402i)T + (4.82 + 1.29i)T^{2} \) |
| 11 | \( 1 + (-1.78 - 1.36i)T + (2.84 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.152 + 0.367i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-1.92 - 3.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.35 - 2.57i)T + (4.91 - 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.986 - 3.68i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.327 - 0.789i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (4.72 + 8.18i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.833 + 6.32i)T + (-35.7 - 9.57i)T^{2} \) |
| 41 | \( 1 + (-2.67 + 2.67i)T - 41iT^{2} \) |
| 43 | \( 1 + (5.88 - 2.43i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-8.48 - 4.89i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.54 + 4.61i)T + (-13.7 - 51.1i)T^{2} \) |
| 59 | \( 1 + (2.96 + 2.27i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (6.11 - 4.69i)T + (15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (-0.798 - 6.06i)T + (-64.7 + 17.3i)T^{2} \) |
| 71 | \( 1 + (-0.641 - 0.641i)T + 71iT^{2} \) |
| 73 | \( 1 + (-13.4 - 3.59i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.67 + 8.09i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.8 - 4.91i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-1.62 + 0.436i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 1.47iT - 97T^{2} \) |
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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
−10.24546832129649241213946556316, −9.542346522762889643606515886072, −9.148528030217673658878186547733, −8.217795055417625377825138827374, −7.19624367442408855725399329542, −6.12993016296288539079123723997, −5.71219258954830797440704771039, −3.66840968009472601503102230035, −2.37967816423169233464985842565, −1.72468708604049641667550990718,
1.06942991051347720697378147062, 2.32700345368918067854559800370, 3.45151353474192802490030115649, 5.04389437154266260526353007431, 6.35449738793525584159261654874, 6.83778854354710387480163920705, 7.931480367030114920703830746924, 9.085062787285995801780196482166, 9.281662221560313035338456966902, 10.34888724291233561422877415413
