L(s) = 1 | − 10.8i·3-s − 26.2·5-s + 86.0·7-s − 36.2·9-s − 114.·11-s + 231. i·13-s + 284. i·15-s − 244.·17-s + (−253. + 256. i)19-s − 932. i·21-s + 269.·23-s + 62.9·25-s − 484. i·27-s + 1.13e3i·29-s + 1.03e3i·31-s + ⋯ |
L(s) = 1 | − 1.20i·3-s − 1.04·5-s + 1.75·7-s − 0.447·9-s − 0.948·11-s + 1.37i·13-s + 1.26i·15-s − 0.845·17-s + (−0.702 + 0.711i)19-s − 2.11i·21-s + 0.508·23-s + 0.100·25-s − 0.664i·27-s + 1.34i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.213905883\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.213905883\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (253. - 256. i)T \) |
good | 3 | \( 1 + 10.8iT - 81T^{2} \) |
| 5 | \( 1 + 26.2T + 625T^{2} \) |
| 7 | \( 1 - 86.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + 114.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 231. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 244.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 269.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.13e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.03e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 302. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.76e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.31e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 835.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 656. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.92e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.57e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.00e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.10e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 7.01e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.26e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.02e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 3.74e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.30e4iT - 8.85e7T^{2} \) |
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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
−11.30629780419568849987571021550, −10.72512722424507063859046570094, −8.861290265838380233236495296595, −8.093193917451140979392147440725, −7.48770378907176778285472731113, −6.62357488782260687601516265415, −5.03704951084203844925359666533, −4.15025912553794428387345385048, −2.23365651691060159290832681857, −1.29016173232681597387000460274,
0.39546035801421531846728426035, 2.48922226232992783993571475006, 4.04240542677564621518770566378, 4.65843444170898927319738730465, 5.54495809552064238509431864507, 7.51653037009047806395072555768, 8.066847021299277737530697181387, 8.967111715171855976818588961528, 10.33811495938867955012790076732, 10.94873318784614794759638088548