L(s) = 1 | − 2-s + (−1.65 − 0.497i)3-s + 4-s + (−1.15 + 1.91i)5-s + (1.65 + 0.497i)6-s − 3.15·7-s − 8-s + (2.50 + 1.64i)9-s + (1.15 − 1.91i)10-s − 2.63·11-s + (−1.65 − 0.497i)12-s + 3.37i·13-s + 3.15·14-s + (2.86 − 2.60i)15-s + 16-s + 0.331i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.957 − 0.286i)3-s + 0.5·4-s + (−0.516 + 0.856i)5-s + (0.677 + 0.202i)6-s − 1.19·7-s − 0.353·8-s + (0.835 + 0.549i)9-s + (0.365 − 0.605i)10-s − 0.795·11-s + (−0.478 − 0.143i)12-s + 0.936i·13-s + 0.844·14-s + (0.740 − 0.672i)15-s + 0.250·16-s + 0.0803i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.229286 - 0.176056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.229286 - 0.176056i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.65 + 0.497i)T \) |
| 5 | \( 1 + (1.15 - 1.91i)T \) |
| 23 | \( 1 + (-2.19 + 4.26i)T \) |
good | 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 - 0.331iT - 17T^{2} \) |
| 19 | \( 1 + 2.06iT - 19T^{2} \) |
| 29 | \( 1 - 6.73iT - 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 7.19T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 13.7iT - 53T^{2} \) |
| 59 | \( 1 + 12.0iT - 59T^{2} \) |
| 61 | \( 1 - 0.685iT - 61T^{2} \) |
| 67 | \( 1 - 0.468T + 67T^{2} \) |
| 71 | \( 1 + 5.57iT - 71T^{2} \) |
| 73 | \( 1 + 7.91iT - 73T^{2} \) |
| 79 | \( 1 - 6.81iT - 79T^{2} \) |
| 83 | \( 1 + 4.16iT - 83T^{2} \) |
| 89 | \( 1 + 7.02T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−10.42764269834178671365339258565, −9.649688640131107069593974050361, −8.553289754677110060335661688601, −7.39076025154464968155203601700, −6.79303543228573708676518013102, −6.25188542672789643439080677344, −4.93029539732734508013875443088, −3.52129141600588403393457199755, −2.30739214168434976802600059298, −0.28744706596872958380551856078,
0.921082072401916545606924602123, 3.01679550368498053403003535629, 4.22768324570724830472445674452, 5.45230565951034153039279100967, 6.08090351145597846539572678122, 7.28510078102795941737503620826, 8.002971056298401507158195846459, 9.125433158042087242109362884239, 9.882051242898330969840377901926, 10.43756104595056143022327607956