L(s) = 1 | − i·3-s − 4-s + i·7-s + i·12-s + 16-s − 2i·17-s + 19-s + 21-s − i·27-s − i·28-s + 29-s − 41-s − i·48-s − 2·51-s − i·53-s + ⋯ |
L(s) = 1 | − i·3-s − 4-s + i·7-s + i·12-s + 16-s − 2i·17-s + 19-s + 21-s − i·27-s − i·28-s + 29-s − 41-s − i·48-s − 2·51-s − i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9081117152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9081117152\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 + T^{2} \) |
| 3 | \( 1 + iT - T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + 2iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.480129619530206138576788579576, −8.740500216264505303932462431537, −7.956948618646354368416497079220, −7.22132896509127137856520811057, −6.35304892711875341882813006434, −5.29047218150297243434544037814, −4.79332800926440087125155738542, −3.36794109633057234360602774170, −2.35822295102411041191841247252, −0.930032885137165132047674498048,
1.31362682131113756051108051552, 3.36457973803491779224947195532, 3.97217197845135319073125238752, 4.61994960548872640396167089395, 5.45405101313080788572919415799, 6.52453096764973979675555908017, 7.63347650989305275420538295227, 8.353602308357405452278563195226, 9.178777683657520965747773266920, 9.945826660555327794197667741707