| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 2.64·7-s + 2.82i·8-s − 17.7i·11-s − 2.58·13-s + 3.74i·14-s + 4.00·16-s − 25.8i·17-s + 20·19-s − 25.1·22-s + 17.7i·23-s + 3.65i·26-s + 5.29·28-s − 11.9i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.377·7-s + 0.353i·8-s − 1.61i·11-s − 0.198·13-s + 0.267i·14-s + 0.250·16-s − 1.52i·17-s + 1.05·19-s − 1.14·22-s + 0.773i·23-s + 0.140i·26-s + 0.188·28-s − 0.410i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8984080230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8984080230\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 11 | \( 1 + 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 2.58T + 169T^{2} \) |
| 17 | \( 1 + 25.8iT - 289T^{2} \) |
| 19 | \( 1 - 20T + 361T^{2} \) |
| 23 | \( 1 - 17.7iT - 529T^{2} \) |
| 29 | \( 1 + 11.9iT - 841T^{2} \) |
| 31 | \( 1 + 17.1T + 961T^{2} \) |
| 37 | \( 1 + 38T + 1.36e3T^{2} \) |
| 41 | \( 1 - 15.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 43.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 85.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 36.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 28.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 120. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 44.4T + 9.40e3T^{2} \) |
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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
−8.151460216137255461288393867790, −7.41153831772188030088142589244, −6.54155721060598632342792114713, −5.49819253585405895090025831550, −5.12833523968328075876854860070, −3.78350560015653480528374144769, −3.24864361750514355881756051539, −2.43482483099051331667303350789, −1.06821097168889736261127895809, −0.22825947710921206303582463852,
1.34503663884165116769059362973, 2.43697609644911746043560273428, 3.67011414951812888896099920246, 4.37555141423981381874894710155, 5.22576691614656178777066626914, 5.97339675127555744289483904806, 6.86105095029017651411208622461, 7.32001429663887075255297559307, 8.085068585657180848689310497750, 8.936849825202705429273747466166
