L(s) = 1 | − 3-s + 2i·5-s + i·7-s + 9-s − 2i·15-s − i·21-s − 3·25-s − 27-s − 2·35-s + 2i·45-s − 49-s + 2·59-s + i·63-s + 3·75-s + 2i·79-s + ⋯ |
L(s) = 1 | − 3-s + 2i·5-s + i·7-s + 9-s − 2i·15-s − i·21-s − 3·25-s − 27-s − 2·35-s + 2i·45-s − 49-s + 2·59-s + i·63-s + 3·75-s + 2i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6825217664\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6825217664\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 2iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 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
−10.12009930328191419208504492812, −9.716622289198695990674773579882, −8.408509846240292794695720495208, −7.34337088562514596514264248598, −6.77216262394216124770656821525, −6.03489072500338395833203482049, −5.40788031447470292434097615155, −4.07232481492294532366636603482, −3.01104115231651150622631176805, −2.05481371925434269775501931639,
0.67180858255211396267284239383, 1.65194193267853584179357215367, 3.86257621589424879599621546664, 4.52948100835376894732925002812, 5.20711583231113957452755109248, 5.99152847418437600599608673085, 7.06875265215283227726168899678, 7.895890369083974894057273419673, 8.715763664452422322788896601995, 9.617482801060288170028143714073