L(s) = 1 | + 1.41i·5-s + 1.41i·11-s + 13-s − 1.41i·17-s − 19-s + 1.41i·23-s − 1.00·25-s − 31-s + 43-s − 49-s + 1.41i·53-s − 2.00·55-s + 1.41i·59-s + 61-s + 1.41i·65-s + ⋯ |
L(s) = 1 | + 1.41i·5-s + 1.41i·11-s + 13-s − 1.41i·17-s − 19-s + 1.41i·23-s − 1.00·25-s − 31-s + 43-s − 49-s + 1.41i·53-s − 2.00·55-s + 1.41i·59-s + 61-s + 1.41i·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.096361973\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.096361973\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.573243202668337204861428860891, −8.987129602501258249038015090450, −7.66356395948192751359859249137, −7.28209321333657416534071415815, −6.54547058168710754302432808774, −5.72350307363656571052338342490, −4.62147982372416173750603605302, −3.67935020901390904550870884872, −2.77944343197474956611119945110, −1.80286987544936973970745573275,
0.826746484795851843203846911133, 2.00262217531102882676359375349, 3.54198703495389148907683921513, 4.18809200065074855229515401565, 5.19582091798717778783100990677, 5.99662915432129118686496093463, 6.55623339616343309362604077027, 8.078500867031149834205514057112, 8.519543626831669197298280336708, 8.800539897295006893726190109579