L(s) = 1 | + (0.271 − 1.38i)2-s + 3-s + (−1.85 − 0.754i)4-s + 0.299i·5-s + (0.271 − 1.38i)6-s + 1.88·7-s + (−1.55 + 2.36i)8-s + 9-s + (0.416 + 0.0814i)10-s + 2.47·11-s + (−1.85 − 0.754i)12-s − 1.99i·13-s + (0.511 − 2.60i)14-s + 0.299i·15-s + (2.86 + 2.79i)16-s + 0.452·17-s + ⋯ |
L(s) = 1 | + (0.192 − 0.981i)2-s + 0.577·3-s + (−0.926 − 0.377i)4-s + 0.134i·5-s + (0.110 − 0.566i)6-s + 0.710·7-s + (−0.548 + 0.836i)8-s + 0.333·9-s + (0.131 + 0.0257i)10-s + 0.744·11-s + (−0.534 − 0.217i)12-s − 0.554i·13-s + (0.136 − 0.697i)14-s + 0.0774i·15-s + (0.715 + 0.698i)16-s + 0.109·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45669 - 1.49137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45669 - 1.49137i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.271 + 1.38i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (2.90 + 7.65i)T \) |
good | 5 | \( 1 - 0.299iT - 5T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 1.99iT - 13T^{2} \) |
| 17 | \( 1 - 0.452T + 17T^{2} \) |
| 19 | \( 1 + 5.21iT - 19T^{2} \) |
| 23 | \( 1 + 0.807iT - 23T^{2} \) |
| 29 | \( 1 - 9.00T + 29T^{2} \) |
| 31 | \( 1 + 8.22T + 31T^{2} \) |
| 37 | \( 1 - 1.31T + 37T^{2} \) |
| 41 | \( 1 + 6.67iT - 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 - 2.51iT - 47T^{2} \) |
| 53 | \( 1 + 0.990iT - 53T^{2} \) |
| 59 | \( 1 - 7.56iT - 59T^{2} \) |
| 61 | \( 1 - 9.06iT - 61T^{2} \) |
| 71 | \( 1 - 0.0862iT - 71T^{2} \) |
| 73 | \( 1 - 1.91T + 73T^{2} \) |
| 79 | \( 1 + 8.30T + 79T^{2} \) |
| 83 | \( 1 - 0.827iT - 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 6.11iT - 97T^{2} \) |
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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
−10.23023455926840830489382647467, −9.099209550757467471442733702516, −8.717215168833003384236030617062, −7.67796437476458184058706255739, −6.55931665797709763227224391495, −5.22629301399780743976567371741, −4.45431048405198661263026161603, −3.36655068944601587254822882515, −2.41197580599474770302676181716, −1.11514113643313398861056736268,
1.50925682611683350861064065841, 3.29159134021812515277685840432, 4.28023792157894009611131912807, 5.08419247557669900828117473125, 6.25126111278760367100692946359, 7.03163111623298162392637195088, 8.001313665584536329375111554535, 8.558294176528166512925063361229, 9.352198679078745397445448338765, 10.17611635523044158850484485223