L(s) = 1 | − 2.67i·2-s + 0.844·4-s + (−10.9 − 2.11i)5-s + 15.4i·7-s − 23.6i·8-s + (−5.65 + 29.3i)10-s + 44.5·11-s − 28.0i·13-s + 41.2·14-s − 56.5·16-s − 92.6i·17-s − 49.5·19-s + (−9.26 − 1.78i)20-s − 119. i·22-s − 0.922i·23-s + ⋯ |
L(s) = 1 | − 0.945i·2-s + 0.105·4-s + (−0.981 − 0.189i)5-s + 0.832i·7-s − 1.04i·8-s + (−0.178 + 0.928i)10-s + 1.22·11-s − 0.597i·13-s + 0.787·14-s − 0.883·16-s − 1.32i·17-s − 0.598·19-s + (−0.103 − 0.0199i)20-s − 1.15i·22-s − 0.00836i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.030096088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.030096088\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (10.9 + 2.11i)T \) |
good | 2 | \( 1 + 2.67iT - 8T^{2} \) |
| 7 | \( 1 - 15.4iT - 343T^{2} \) |
| 11 | \( 1 - 44.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 92.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 49.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.922iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 57.8iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 0.591iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 598. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 146. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 193.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 566.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 355. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 320.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 636. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 287.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 285. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 331.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3iT - 9.12e5T^{2} \) |
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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
−10.64073732857716285352217139484, −9.405506588550675091967430098040, −8.827545726439097085919068662890, −7.52216581403598501268334354307, −6.68761016859541030299849957391, −5.33054291568289356916040739324, −3.98474126591307496236874528241, −3.15058157943089793686568327926, −1.82257818881620414362284806029, −0.33268622890634167881545057874,
1.68055757000529195869479657162, 3.63598085636196034220391261460, 4.36946036278885485483575446100, 5.90060029210838188348025619731, 6.81976471063727728192260424333, 7.38522395929396702602217737132, 8.318548311475368850399573836176, 9.185861532411412571817194660401, 10.68416717340197256066558379746, 11.20003447905153756847293938049