| L(s) = 1 | + (−1.53 − 1.28i)2-s + (−0.598 + 0.217i)3-s + (0.694 + 3.93i)4-s + (3.08 − 17.4i)5-s + (1.19 + 0.435i)6-s + (−11.6 − 20.0i)7-s + (4.00 − 6.92i)8-s + (−20.3 + 17.0i)9-s + (−27.1 + 22.8i)10-s + (20.1 − 34.9i)11-s + (−1.27 − 2.20i)12-s + (58.0 + 21.1i)13-s + (−8.05 + 45.7i)14-s + (1.96 + 11.1i)15-s + (−15.0 + 5.47i)16-s + (54.7 + 45.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.115 + 0.0418i)3-s + (0.0868 + 0.492i)4-s + (0.275 − 1.56i)5-s + (0.0813 + 0.0296i)6-s + (−0.626 − 1.08i)7-s + (0.176 − 0.306i)8-s + (−0.754 + 0.633i)9-s + (−0.860 + 0.721i)10-s + (0.553 − 0.958i)11-s + (−0.0306 − 0.0530i)12-s + (1.23 + 0.450i)13-s + (−0.153 + 0.872i)14-s + (0.0337 + 0.191i)15-s + (−0.234 + 0.0855i)16-s + (0.780 + 0.655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.523903 - 0.714153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.523903 - 0.714153i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.53 + 1.28i)T \) |
| 19 | \( 1 + (81.9 + 12.0i)T \) |
| good | 3 | \( 1 + (0.598 - 0.217i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (-3.08 + 17.4i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (11.6 + 20.0i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-20.1 + 34.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-58.0 - 21.1i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-54.7 - 45.9i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-22.2 - 126. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-183. + 153. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (18.7 + 32.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-110. + 40.1i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-30.8 + 174. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (1.88 - 1.57i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-18.5 - 105. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-188. - 158. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-108. - 616. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (139. - 117. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-174. + 990. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (366. - 133. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (823. - 299. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-216. - 374. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-361. - 131. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-718. - 603. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.13716010694294987860344823765, −13.76431740685565688408507843233, −13.16694907882178225405961057758, −11.70710185557951257731138699225, −10.50229975384971026515759001655, −9.066330319939339539889880667562, −8.183612657614988546930529004898, −5.99579535340150774007743874337, −3.97246093674577664692548600794, −0.992084779026225593661686827008,
2.89962813214858998986751737993, 6.03661871006736444433212470451, 6.73965298329446763824487322346, 8.640183247360709844805170939186, 9.896309054132631218260684126319, 11.09211984989449277335053019379, 12.44995192715105337348630184054, 14.38240246430360466163205237081, 14.91718617655000880285930751836, 16.06944505250110581366049205796
