L(s) = 1 | − 5-s + 4.35i·7-s − 3·9-s + 4.35i·11-s + 7·17-s − 4.35i·19-s + 8.71i·23-s − 4·25-s − 4.35i·35-s − 13.0i·43-s + 3·45-s + 4.35i·47-s − 12.0·49-s − 4.35i·55-s + 15·61-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.64i·7-s − 9-s + 1.31i·11-s + 1.69·17-s − 0.999i·19-s + 1.81i·23-s − 0.800·25-s − 0.736i·35-s − 1.99i·43-s + 0.447·45-s + 0.635i·47-s − 1.71·49-s − 0.587i·55-s + 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.709639 + 0.709639i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.709639 + 0.709639i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 4.35iT - 7T^{2} \) |
| 11 | \( 1 - 4.35iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 23 | \( 1 - 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 13.0iT - 43T^{2} \) |
| 47 | \( 1 - 4.35iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.93844245095129171959488032957, −11.37058711974801084999360835271, −9.865224886415717308099031813263, −9.169588155260978087686422243889, −8.176959562550778912654720271459, −7.25110734562622309623475704069, −5.76173891115666305871457994825, −5.17400013551212808950091505439, −3.45263935766864084060798214387, −2.20554819837438602305096567011,
0.75067994626145005201323780568, 3.18985105245041001800734306256, 4.04036788852722839709519649446, 5.54429051267460212401686320132, 6.57397978870308106344754277290, 7.935279911670733220867378369606, 8.246007557642040916600789169367, 9.810156314211485736748144542945, 10.64039820529012766146601931497, 11.36424100874035080602044629917