L(s) = 1 | + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (3 − 5.19i)11-s + (0.5 + 0.866i)13-s − 19-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)25-s + (3 − 5.19i)29-s + (−4 − 6.92i)31-s + 0.999·35-s − 7·37-s + (−3 − 5.19i)41-s + (2 − 3.46i)43-s + (6 − 10.3i)47-s + (3 + 5.19i)49-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (0.188 − 0.327i)7-s + (0.904 − 1.56i)11-s + (0.138 + 0.240i)13-s − 0.229·19-s + (0.625 + 1.08i)23-s + (−0.0999 + 0.173i)25-s + (0.557 − 0.964i)29-s + (−0.718 − 1.24i)31-s + 0.169·35-s − 1.15·37-s + (−0.468 − 0.811i)41-s + (0.304 − 0.528i)43-s + (0.875 − 1.51i)47-s + (0.428 + 0.742i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.864991599\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.864991599\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.117103165356682067205774154988, −8.707507576078659200805731718439, −7.64949770793006575000847689344, −6.88927875059497253503524900796, −6.03255574087117669283223451061, −5.39609020589636824390682936124, −4.01050746590354684390878290993, −3.46317722579935583968584931004, −2.16232661081501294189568825430, −0.817107985060873369225355709621,
1.29058377336987598391147609536, 2.28209736958848734266206952127, 3.56683067622703208294517785351, 4.68973545747129347315240438732, 5.13815789471283498099003027906, 6.43878483426655247084553947299, 6.93452850895556673355372459868, 7.948433025806952808223705155257, 8.896083244660351379071625984535, 9.275862770202948730539458817071