L(s) = 1 | + (0.541 + 1.30i)2-s + (1.30 + 1.13i)3-s + (−1.41 + 1.41i)4-s + (−2.10 − 0.765i)5-s + (−0.778 + 2.32i)6-s + 2.27·7-s + (−2.61 − 1.08i)8-s + (0.414 + 2.97i)9-s + (−0.137 − 3.15i)10-s − 4.20i·11-s + (−3.45 + 0.239i)12-s + 3.21·13-s + (1.23 + 2.97i)14-s + (−1.87 − 3.38i)15-s − 4i·16-s + 1.53·17-s + ⋯ |
L(s) = 1 | + (0.382 + 0.923i)2-s + (0.754 + 0.656i)3-s + (−0.707 + 0.707i)4-s + (−0.939 − 0.342i)5-s + (−0.317 + 0.948i)6-s + 0.859·7-s + (−0.923 − 0.382i)8-s + (0.138 + 0.990i)9-s + (−0.0433 − 0.999i)10-s − 1.26i·11-s + (−0.997 + 0.0692i)12-s + 0.891·13-s + (0.328 + 0.794i)14-s + (−0.484 − 0.875i)15-s − i·16-s + 0.371·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.902873 + 1.01073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.902873 + 1.01073i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.541 - 1.30i)T \) |
| 3 | \( 1 + (-1.30 - 1.13i)T \) |
| 5 | \( 1 + (2.10 + 0.765i)T \) |
good | 7 | \( 1 - 2.27T + 7T^{2} \) |
| 11 | \( 1 + 4.20iT - 11T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 1.08iT - 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 + 6.82iT - 31T^{2} \) |
| 37 | \( 1 + 7.76T + 37T^{2} \) |
| 41 | \( 1 - 2.46iT - 41T^{2} \) |
| 43 | \( 1 + 8.70iT - 43T^{2} \) |
| 47 | \( 1 + 1.08iT - 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.20iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 2.27iT - 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 4.54iT - 73T^{2} \) |
| 79 | \( 1 + 0.485iT - 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 8.40iT - 89T^{2} \) |
| 97 | \( 1 - 10.9iT - 97T^{2} \) |
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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
−13.94802213075734169727285895460, −13.15249246524076402143237062309, −11.72680862671100206244647533593, −10.71329372075851512599534327716, −8.867759310216032457774283179868, −8.431243802206786016026014049385, −7.52931602237207382503055771970, −5.72492333139314635187667247031, −4.41687383026674587875445185159, −3.48502519311608542932796251612,
1.82041611211165762078027645365, 3.46866357297740168708394365089, 4.65946841741135282302873134696, 6.66019287383440306153192978620, 7.982847446017719924934080919433, 8.843976315282772390421537538892, 10.29893566510207950517566437808, 11.34142946486021755352006694622, 12.27362537179254073488571612798, 12.97216082706330350974783086286