L(s) = 1 | + 2-s + 3-s + 4-s − 2.41·5-s + 6-s + 7-s + 8-s + 9-s − 2.41·10-s + 3.24·11-s + 12-s + 14-s − 2.41·15-s + 16-s + 1.80·17-s + 18-s − 2.73·19-s − 2.41·20-s + 21-s + 3.24·22-s − 6.58·23-s + 24-s + 0.832·25-s + 27-s + 28-s + 6.22·29-s − 2.41·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.08·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.763·10-s + 0.979·11-s + 0.288·12-s + 0.267·14-s − 0.623·15-s + 0.250·16-s + 0.437·17-s + 0.235·18-s − 0.628·19-s − 0.540·20-s + 0.218·21-s + 0.692·22-s − 1.37·23-s + 0.204·24-s + 0.166·25-s + 0.192·27-s + 0.188·28-s + 1.15·29-s − 0.440·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.761726962\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.761726962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2.41T + 5T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 - 6.22T + 29T^{2} \) |
| 31 | \( 1 - 2.72T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 - 9.15T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 - 9.77T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 + 1.50T + 71T^{2} \) |
| 73 | \( 1 + 0.786T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.82055953949008977692079660702, −7.42811044961945008797415373243, −6.39809700872875585934267916066, −6.00079425390356674788822760075, −4.74538642623872064068852892380, −4.23588211455717428837526343113, −3.77057729435417352303869476737, −2.89115902594235993024331458770, −2.00071980547430035999588849957, −0.881392460209941472974825371294,
0.881392460209941472974825371294, 2.00071980547430035999588849957, 2.89115902594235993024331458770, 3.77057729435417352303869476737, 4.23588211455717428837526343113, 4.74538642623872064068852892380, 6.00079425390356674788822760075, 6.39809700872875585934267916066, 7.42811044961945008797415373243, 7.82055953949008977692079660702