| L(s) = 1 | + (0.996 + 0.0871i)2-s + (1.39 + 1.03i)3-s + (0.984 + 0.173i)4-s + (0.209 − 2.22i)5-s + (1.29 + 1.14i)6-s + (−0.734 − 0.514i)7-s + (0.965 + 0.258i)8-s + (0.877 + 2.86i)9-s + (0.402 − 2.19i)10-s + (0.457 − 1.25i)11-s + (1.19 + 1.25i)12-s + (0.0942 + 1.07i)13-s + (−0.687 − 0.576i)14-s + (2.58 − 2.88i)15-s + (0.939 + 0.342i)16-s + (−3.60 + 0.965i)17-s + ⋯ |
| L(s) = 1 | + (0.704 + 0.0616i)2-s + (0.803 + 0.594i)3-s + (0.492 + 0.0868i)4-s + (0.0935 − 0.995i)5-s + (0.529 + 0.468i)6-s + (−0.277 − 0.194i)7-s + (0.341 + 0.0915i)8-s + (0.292 + 0.956i)9-s + (0.127 − 0.695i)10-s + (0.137 − 0.378i)11-s + (0.344 + 0.362i)12-s + (0.0261 + 0.298i)13-s + (−0.183 − 0.154i)14-s + (0.667 − 0.744i)15-s + (0.234 + 0.0855i)16-s + (−0.873 + 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28880 + 0.235810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28880 + 0.235810i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.996 - 0.0871i)T \) |
| 3 | \( 1 + (-1.39 - 1.03i)T \) |
| 5 | \( 1 + (-0.209 + 2.22i)T \) |
| good | 7 | \( 1 + (0.734 + 0.514i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.457 + 1.25i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.0942 - 1.07i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (3.60 - 0.965i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.47 - 2.58i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.187 + 0.268i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-3.73 + 3.13i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.54 - 8.76i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.74 + 10.2i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.945 + 1.12i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.475 - 1.02i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.63 + 6.61i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (2.08 - 2.08i)T - 53iT^{2} \) |
| 59 | \( 1 + (-11.7 + 4.26i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 5.97i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.51 - 0.832i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (2.95 + 1.70i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.15 - 4.30i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-10.7 - 12.8i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.298 - 3.40i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (4.47 + 7.75i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.5 - 4.89i)T + (62.3 - 74.3i)T^{2} \) |
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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
−12.23696865784585470497433572941, −10.96178020301954002320066109526, −10.07634715552393707596536724608, −8.887940282757231833281397622421, −8.360088197335834927399633968279, −6.96576773377084504357485544867, −5.64689432493003260203332167129, −4.49144603179536300034034028346, −3.72561373940828356080186509465, −2.09615557795041720632034111843,
2.19477824675009042999125123172, 3.10668517636628742953816606534, 4.39471956951269662843340781322, 6.17677988142322811027977608777, 6.79905974613112310925238194445, 7.77129222707662082190765293170, 8.994834227449003087018990870745, 10.06481471121736907120718361981, 11.10199973369396728655873136211, 12.05796888585558378168061822348
