L(s) = 1 | + (−0.707 + 0.707i)5-s − i·7-s + (−0.707 + 0.707i)11-s + (1 − i)13-s − 1.41i·17-s − 31-s + (0.707 + 0.707i)35-s + (−1 − i)43-s − 1.41i·47-s + (0.707 − 0.707i)53-s − 1.00i·55-s + 1.41i·65-s + (1 − i)67-s + 1.41·71-s − i·73-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)5-s − i·7-s + (−0.707 + 0.707i)11-s + (1 − i)13-s − 1.41i·17-s − 31-s + (0.707 + 0.707i)35-s + (−1 − i)43-s − 1.41i·47-s + (0.707 − 0.707i)53-s − 1.00i·55-s + 1.41i·65-s + (1 − i)67-s + 1.41·71-s − i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9182783265\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9182783265\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.458227521300123556931706190028, −7.70263832172441593185996216092, −7.23418904936557053699583099484, −6.69296658059264737505827699050, −5.48714092260932165488315343289, −4.85049348859462938965811847660, −3.69074829494370342818654439173, −3.34963672849255140260452175914, −2.15059653316745796739239460530, −0.58022039473664293180771369122,
1.31873431063958502237368471649, 2.42081441158556368020199952387, 3.57164705183215532076690067792, 4.20549152254513557178226552028, 5.17149725837332307746917214641, 5.91461776435213712864891445784, 6.49533560280140055355697855853, 7.66934353353795388038504636646, 8.420370832209908210860145789093, 8.637942168494806061901634581335