L(s) = 1 | + 2-s − 5-s − 1.41i·7-s − 8-s − 10-s − 11-s − 1.41i·14-s − 16-s − 1.41i·19-s − 22-s − 1.41i·29-s − 31-s + 1.41i·35-s + 37-s − 1.41i·38-s + ⋯ |
L(s) = 1 | + 2-s − 5-s − 1.41i·7-s − 8-s − 10-s − 11-s − 1.41i·14-s − 16-s − 1.41i·19-s − 22-s − 1.41i·29-s − 31-s + 1.41i·35-s + 37-s − 1.41i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7964915119\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7964915119\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 173 | \( 1 + T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−9.422022201807035411463585512867, −8.355766052940446791035915645614, −7.62529435469117780657530363194, −7.00076166694260413787766874734, −5.94528301480961380371396082373, −4.88426456119280864751110425569, −4.30843548309651575955451710462, −3.63276283596606777575182113673, −2.64865689306291155330659417459, −0.43944716434828770620498518772,
2.22448097382365254428889267730, 3.27685611069167612332900559098, 3.93420004713447177983923367226, 5.14493191129841691554501260915, 5.46370903452346113168792723288, 6.41356676929251892677435884004, 7.58596541082838204627691527792, 8.345523152203410392106412646763, 8.922280636046533671087612600388, 9.882100179514305725913547209887