L(s) = 1 | + (0.840 + 1.13i)2-s + (−0.588 + 1.91i)4-s + 2.67i·5-s + (−0.370 − 2.61i)7-s + (−2.66 + 0.935i)8-s + (−3.04 + 2.25i)10-s + 1.39i·11-s + 3.08i·13-s + (2.66 − 2.62i)14-s + (−3.30 − 2.25i)16-s + 5.83i·17-s − 1.91·19-s + (−5.12 − 1.57i)20-s + (−1.58 + 1.17i)22-s − 1.39i·23-s + ⋯ |
L(s) = 1 | + (0.593 + 0.804i)2-s + (−0.294 + 0.955i)4-s + 1.19i·5-s + (−0.140 − 0.990i)7-s + (−0.943 + 0.330i)8-s + (−0.963 + 0.711i)10-s + 0.421i·11-s + 0.855i·13-s + (0.713 − 0.700i)14-s + (−0.826 − 0.562i)16-s + 1.41i·17-s − 0.440·19-s + (−1.14 − 0.352i)20-s + (−0.338 + 0.250i)22-s − 0.291i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.122635 + 1.54656i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.122635 + 1.54656i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.840 - 1.13i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.370 + 2.61i)T \) |
good | 5 | \( 1 - 2.67iT - 5T^{2} \) |
| 11 | \( 1 - 1.39iT - 11T^{2} \) |
| 13 | \( 1 - 3.08iT - 13T^{2} \) |
| 17 | \( 1 - 5.83iT - 17T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 23 | \( 1 + 1.39iT - 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 + 9.53T + 31T^{2} \) |
| 37 | \( 1 - 6.83T + 37T^{2} \) |
| 41 | \( 1 - 0.693iT - 41T^{2} \) |
| 43 | \( 1 + 0.678iT - 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 12.8iT - 61T^{2} \) |
| 67 | \( 1 - 3.08iT - 67T^{2} \) |
| 71 | \( 1 - 9.79iT - 71T^{2} \) |
| 73 | \( 1 - 5.91iT - 73T^{2} \) |
| 79 | \( 1 - 12.0iT - 79T^{2} \) |
| 83 | \( 1 + 5.77T + 83T^{2} \) |
| 89 | \( 1 + 6.65iT - 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−10.82909440109177000707576024795, −9.955395199557700262069264958210, −8.895036368032323994941182320279, −7.81585189976754459227601488173, −7.04729840536581679639101147968, −6.59169487098454443883793303962, −5.59189057427683320722801024358, −4.13131161275769672827042220366, −3.73570668547176715776893007636, −2.27169606958766619220203513245,
0.63699145618300570659484359183, 2.15718524570644720074794704573, 3.26163825773620036708597661259, 4.47822548910558256735080728535, 5.47278376140292331831087165482, 5.76380487775306948958824863191, 7.34941440394538462765464423246, 8.692840457910745797915411833986, 9.080277763690591492048958151587, 9.899315253560473055302655472373