L(s) = 1 | + (−1 − i)5-s − 9-s − 13-s + 2i·17-s + i·25-s + 2·29-s + (−1 + i)37-s + (1 + i)41-s + (1 + i)45-s + i·49-s + (1 + i)65-s + (1 − i)73-s + 81-s + (2 − 2i)85-s + (−1 + i)89-s + ⋯ |
L(s) = 1 | + (−1 − i)5-s − 9-s − 13-s + 2i·17-s + i·25-s + 2·29-s + (−1 + i)37-s + (1 + i)41-s + (1 + i)45-s + i·49-s + (1 + i)65-s + (1 − i)73-s + 81-s + (2 − 2i)85-s + (−1 + i)89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5753204244\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5753204244\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + (1 + i)T + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (1 - i)T - iT^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−8.683566104238731047272374834983, −8.242141702300957214717329453649, −7.81520521953101285156699409397, −6.66650127388659503924260990196, −5.94567037741323216314380736925, −4.97240062320504242813866530914, −4.43827007065705420078543802342, −3.53349567947064005915898373053, −2.56133958947634837638968183931, −1.18954866088576765547196896422,
0.37409227542500695295941811801, 2.53719421717224670509687213378, 2.88168046444037623653313610715, 3.86651587894082261924268524192, 4.86653243440099295009878688818, 5.53177382797696908474011046011, 6.72916288767080884441788063230, 7.12030323217604796249396735785, 7.80401936377682491840190525021, 8.582985594314454595550991724989