L(s) = 1 | + 8·4-s + 20i·7-s + 70i·13-s + 64·16-s − 56·19-s + 160i·28-s + 308·31-s + 110i·37-s + 520i·43-s − 57·49-s + 560i·52-s + 182·61-s + 512·64-s − 880i·67-s − 1.19e3i·73-s + ⋯ |
L(s) = 1 | + 4-s + 1.07i·7-s + 1.49i·13-s + 16-s − 0.676·19-s + 1.07i·28-s + 1.78·31-s + 0.488i·37-s + 1.84i·43-s − 0.166·49-s + 1.49i·52-s + 0.382·61-s + 64-s − 1.60i·67-s − 1.90i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.80270 + 1.11412i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80270 + 1.11412i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 8T^{2} \) |
| 7 | \( 1 - 20iT - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 70iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 56T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 308T + 2.97e4T^{2} \) |
| 37 | \( 1 - 110iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 520iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 182T + 2.26e5T^{2} \) |
| 67 | \( 1 + 880iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.19e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 884T + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3iT - 9.12e5T^{2} \) |
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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
−11.83317629619339750275411306096, −11.27106770442625783710335372703, −10.05937957131318552609303742528, −9.008535564018519969191632595058, −8.001600599237077787702217421772, −6.69011148840149696759225328526, −6.05506834780535245079356182681, −4.57589828502556953704423015645, −2.87111924768286360602338227764, −1.77300373682735636680133638761,
0.887581425520046305461421266470, 2.62208392113583812603295536198, 3.91851140829057833404866719228, 5.49000046502699681644352355077, 6.64584941265074391384830175252, 7.52623313079840970827005365401, 8.418711145578016408013338854172, 10.18413032330575647567067529043, 10.48643982641229224489160539567, 11.54220132617010680369072812269