L(s) = 1 | + (0.841 − 0.540i)2-s + (0.936 + 0.349i)3-s + (0.415 − 0.909i)4-s + (0.989 − 0.142i)5-s + (0.977 − 0.212i)6-s + (−0.800 − 0.599i)7-s + (−0.142 − 0.989i)8-s + (0.755 + 0.654i)9-s + (0.755 − 0.654i)10-s + (0.142 − 0.989i)11-s + (0.707 − 0.707i)12-s + (−0.349 + 0.936i)13-s + (−0.997 − 0.0713i)14-s + (0.977 + 0.212i)15-s + (−0.654 − 0.755i)16-s + (−0.540 + 0.841i)17-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (0.936 + 0.349i)3-s + (0.415 − 0.909i)4-s + (0.989 − 0.142i)5-s + (0.977 − 0.212i)6-s + (−0.800 − 0.599i)7-s + (−0.142 − 0.989i)8-s + (0.755 + 0.654i)9-s + (0.755 − 0.654i)10-s + (0.142 − 0.989i)11-s + (0.707 − 0.707i)12-s + (−0.349 + 0.936i)13-s + (−0.997 − 0.0713i)14-s + (0.977 + 0.212i)15-s + (−0.654 − 0.755i)16-s + (−0.540 + 0.841i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.463697513 - 1.797968156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.463697513 - 1.797968156i\) |
\(L(1)\) |
\(\approx\) |
\(2.276259578 - 0.7843541199i\) |
\(L(1)\) |
\(\approx\) |
\(2.276259578 - 0.7843541199i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.936 + 0.349i)T \) |
| 5 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (-0.800 - 0.599i)T \) |
| 11 | \( 1 + (0.142 - 0.989i)T \) |
| 13 | \( 1 + (-0.349 + 0.936i)T \) |
| 17 | \( 1 + (-0.540 + 0.841i)T \) |
| 19 | \( 1 + (0.997 - 0.0713i)T \) |
| 23 | \( 1 + (0.0713 + 0.997i)T \) |
| 29 | \( 1 + (-0.599 + 0.800i)T \) |
| 31 | \( 1 + (0.0713 - 0.997i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.349 + 0.936i)T \) |
| 43 | \( 1 + (-0.599 - 0.800i)T \) |
| 47 | \( 1 + (0.909 + 0.415i)T \) |
| 53 | \( 1 + (-0.909 + 0.415i)T \) |
| 59 | \( 1 + (-0.936 + 0.349i)T \) |
| 61 | \( 1 + (0.877 - 0.479i)T \) |
| 67 | \( 1 + (0.415 + 0.909i)T \) |
| 71 | \( 1 + (-0.989 - 0.142i)T \) |
| 73 | \( 1 + (0.654 + 0.755i)T \) |
| 79 | \( 1 + (-0.755 + 0.654i)T \) |
| 83 | \( 1 + (-0.977 + 0.212i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.60803589162253240077969507533, −29.68451518586494478802623956882, −28.72102895435760222472719103929, −26.73208361144343568773922543912, −25.79876726308858949400348904351, −25.01103547374932288111856001229, −24.58491939968854068618916322478, −22.769682911829406046039438868137, −22.15779067808784043372123917019, −20.80294152465218837258121294995, −20.07936752969461277092393840897, −18.42165208649375848754633092739, −17.47504308041238509056356741418, −15.876968740087843249676268317, −14.97213783191306750871578568639, −13.94235251742844296183602099071, −13.006958094943559444087796415515, −12.21237897910845746182384737421, −9.97980022586496227060389849736, −8.91251339746665269770204163537, −7.37222286296006266923652818445, −6.40860899345282390891951849941, −5.01110327505211871421037793520, −3.15995025616911233544008203539, −2.252652378668407051182175029144,
1.57895982574319430186703516706, 3.00634602093942234076593350373, 4.10500690463073644551772715232, 5.67557616116132188326525619700, 7.01386302884470822632037020535, 9.1193709308205382350304809813, 9.87318269347149548187761437214, 11.06122304402992719774488199818, 12.859787061085113803508860051683, 13.6781232734543255657533241328, 14.27857135138546642838469972077, 15.72643354112934979280962350857, 16.80115367933412773334065223572, 18.76262602477612752501721394438, 19.64865285793402262690709605261, 20.57155263464562988021089937408, 21.693016898112957525780985218423, 22.122610364190419256864431459158, 23.86412111347439629317755548995, 24.69719419750218586837367319372, 25.87968161267210525501825884392, 26.7493193177683616763746006612, 28.35035679762145614686281022134, 29.36416248094124887944802075530, 30.02298068693498530415945669761