L(s) = 1 | + (−0.5 + 0.866i)5-s + (2 + 3.46i)7-s + (−2 − 3.46i)11-s + (1 − 1.73i)13-s + 2·17-s + 4·19-s + (−2 + 3.46i)23-s + (−0.499 − 0.866i)25-s + (1 + 1.73i)29-s + (4 − 6.92i)31-s − 3.99·35-s + 6·37-s + (3 − 5.19i)41-s + (4 + 6.92i)43-s + (−2 − 3.46i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.755 + 1.30i)7-s + (−0.603 − 1.04i)11-s + (0.277 − 0.480i)13-s + 0.485·17-s + 0.917·19-s + (−0.417 + 0.722i)23-s + (−0.0999 − 0.173i)25-s + (0.185 + 0.321i)29-s + (0.718 − 1.24i)31-s − 0.676·35-s + 0.986·37-s + (0.468 − 0.811i)41-s + (0.609 + 1.05i)43-s + (−0.291 − 0.505i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.979216631\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.979216631\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8 - 13.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.607775709437041602413722924695, −7.963075513021214427898954049499, −7.55791260785049701360181746942, −6.24450688203629144218535554065, −5.65567464688184135549705525501, −5.18120220181573496801746430151, −3.96394779183587688187157266762, −3.00087529291521224842535752278, −2.36936645781357900319827397737, −0.971950573948978149389944219909,
0.799323112237009899087073249450, 1.73186032262769934670712188038, 2.97768098757120242443489721192, 4.17352343707517017505178362686, 4.53974100949565747037044229012, 5.33975269808058419047303791955, 6.43646634818537288565510981220, 7.33388862660499371511840652318, 7.69466489956796322098432985878, 8.417825159260472172621727280129