L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)5-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·10-s + (−0.707 + 0.707i)11-s + (−1 + i)13-s + (0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (−0.707 − 0.707i)20-s + 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)5-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·10-s + (−0.707 + 0.707i)11-s + (−1 + i)13-s + (0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (−0.707 − 0.707i)20-s + 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.216265391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216265391\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 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
−11.55293476886667045899638210271, −10.22037721964235543898145333497, −9.478054472216611856689181416419, −8.976497423871855767675419856033, −7.36313114954289662332727941628, −6.17306625928658912296584275359, −5.07640945539313864995736517812, −4.71388316553114415104045157819, −2.79685177606582793573701173399, −1.92571696198638370433387376998,
2.59097517581282865672808478937, 3.64572877319623334347323177999, 4.95121193538855816691733734501, 5.94129394461114701075032964697, 6.74305470865466984333616645250, 7.73768074330630743817850565657, 8.423625649063914643811153903773, 10.07811484228286977577933485556, 10.46200996594786302416744238994, 11.58706984603199617260118832604