| L(s) = 1 | + 4i·2-s − 16·4-s + (−10 + 55i)5-s + 49i·7-s − 64i·8-s + (−220 − 40i)10-s + 175·11-s − 999i·13-s − 196·14-s + 256·16-s + 1.83e3i·17-s + 1.30e3·19-s + (160 − 880i)20-s + 700i·22-s + 4.19e3i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.178 + 0.983i)5-s + 0.377i·7-s − 0.353i·8-s + (−0.695 − 0.126i)10-s + 0.436·11-s − 1.63i·13-s − 0.267·14-s + 0.250·16-s + 1.53i·17-s + 0.831·19-s + (0.0894 − 0.491i)20-s + 0.308i·22-s + 1.65i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3791360521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3791360521\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (10 - 55i)T \) |
| 7 | \( 1 - 49iT \) |
| good | 11 | \( 1 - 175T + 1.61e5T^{2} \) |
| 13 | \( 1 + 999iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.83e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.30e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.19e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 981T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 578iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.95e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.02e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.56e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.98e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.35e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.30e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.37e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.41e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.32e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.85e4iT - 8.58e9T^{2} \) |
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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
−10.32649329937126826631450611010, −9.622367024040758668078703613659, −8.450745758242235071371774879490, −7.75625838617911555206903034974, −6.96780557869018409829727818627, −5.91348945131677205917222265841, −5.38017176341056693083815532636, −3.77833282633999984408914891542, −3.13729642734224738048804065976, −1.54670497090417014251526066946,
0.088866571507929971784595580389, 1.09974638512053633127854037312, 2.13435792778833599794957645534, 3.54721993927074358501300979625, 4.50300841856144636563315184418, 5.11191410971818362560157959415, 6.56698017196206863293800743598, 7.47860949992633766863685573132, 8.675682570444741274093642801523, 9.230230130493563012245875237305
