L(s) = 1 | − 1.73·3-s + 4.29i·5-s − 2.75i·7-s + 2.99·9-s − 13.7·11-s − 14.5i·13-s − 7.43i·15-s + 22.8·17-s + 16.0·19-s + 4.77i·21-s + 17.1i·23-s + 6.57·25-s − 5.19·27-s + 21.8i·29-s − 38.6i·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.858i·5-s − 0.393i·7-s + 0.333·9-s − 1.25·11-s − 1.11i·13-s − 0.495i·15-s + 1.34·17-s + 0.844·19-s + 0.227i·21-s + 0.743i·23-s + 0.262·25-s − 0.192·27-s + 0.754i·29-s − 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.330826638\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.330826638\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
good | 5 | \( 1 - 4.29iT - 25T^{2} \) |
| 7 | \( 1 + 2.75iT - 49T^{2} \) |
| 11 | \( 1 + 13.7T + 121T^{2} \) |
| 13 | \( 1 + 14.5iT - 169T^{2} \) |
| 17 | \( 1 - 22.8T + 289T^{2} \) |
| 19 | \( 1 - 16.0T + 361T^{2} \) |
| 23 | \( 1 - 17.1iT - 529T^{2} \) |
| 29 | \( 1 - 21.8iT - 841T^{2} \) |
| 31 | \( 1 + 38.6iT - 961T^{2} \) |
| 37 | \( 1 - 66.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 23.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 47.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 14.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 65.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 40.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 74.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 122. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 144.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 22.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 122.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88.3T + 9.40e3T^{2} \) |
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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
−10.13059080481080576061272939566, −9.926374723428371034555719824089, −8.226868383078025613470532232385, −7.58361756904854907260170529982, −6.83613957767792144059355300242, −5.60798924774652785026678087116, −5.15794937155369604000640910222, −3.56016388635469897932883566823, −2.77106190314119973485514876083, −0.955150000929451931770958047171,
0.65636016972801211036613603086, 2.08337619561973575953699000735, 3.55984974406104938023703563265, 4.97987933480995969895638162981, 5.25606523083654023790122620861, 6.41084555711524146085357847157, 7.47204404064248514401713202001, 8.318900932360564046967097748056, 9.200753725598255187368066410306, 10.02295006994927516896994216248