| L(s) = 1 | − 8.05i·5-s − 20.0i·7-s + 14.7·11-s − 12.7·13-s + 58.0i·17-s − 10.9i·19-s − 74.3·23-s + 60.1·25-s − 144. i·29-s − 181. i·31-s − 161.·35-s − 99.7·37-s − 249. i·41-s − 201. i·43-s + 416.·47-s + ⋯ |
| L(s) = 1 | − 0.720i·5-s − 1.08i·7-s + 0.403·11-s − 0.272·13-s + 0.827i·17-s − 0.132i·19-s − 0.674·23-s + 0.481·25-s − 0.925i·29-s − 1.05i·31-s − 0.779·35-s − 0.443·37-s − 0.951i·41-s − 0.715i·43-s + 1.29·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.070371595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.070371595\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.05iT - 125T^{2} \) |
| 7 | \( 1 + 20.0iT - 343T^{2} \) |
| 11 | \( 1 - 14.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 10.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 74.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 181. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 99.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 201. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 416.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 420. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 266.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 199.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 55.6iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 704.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 751. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 746.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 575.T + 9.12e5T^{2} \) |
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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
−8.555712862724816860264909000681, −7.78901653206710997113617150428, −7.04655018017801432926095396177, −6.15259075569083888309064877692, −5.25069913025199676929392189427, −4.20827008045610269553799127275, −3.80834101113565245306761466865, −2.29874643781246690470358391806, −1.17478958866731295877258601390, −0.23832178500955125029555537601,
1.41466864337703951271330948580, 2.61103203775847593161181404856, 3.18969878606222644037214927063, 4.46652442603248739115936110652, 5.36347585343692407375937029484, 6.18131320580618230998551330040, 6.94227264633447228096096098183, 7.69107664017674899234112225948, 8.763747999080782901227902650303, 9.191973891274258728537348835421
