L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (0.707 + 0.707i)5-s − 0.765i·7-s + (−0.382 + 0.923i)8-s − i·9-s + (−0.923 − 0.382i)10-s + (−0.707 − 0.707i)11-s + (1.30 − 1.30i)13-s + (0.292 + 0.707i)14-s − i·16-s − 0.765·17-s + (0.382 + 0.923i)18-s + 20-s + (0.923 + 0.382i)22-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (0.707 + 0.707i)5-s − 0.765i·7-s + (−0.382 + 0.923i)8-s − i·9-s + (−0.923 − 0.382i)10-s + (−0.707 − 0.707i)11-s + (1.30 − 1.30i)13-s + (0.292 + 0.707i)14-s − i·16-s − 0.765·17-s + (0.382 + 0.923i)18-s + 20-s + (0.923 + 0.382i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7248004136\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7248004136\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 - 0.382i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 + 0.765iT - T^{2} \) |
| 13 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 17 | \( 1 + 0.765T + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-1 - i)T + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.84iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.38443878470421074697145246282, −9.415315973378110133371533119747, −8.669885893224801684468793962773, −7.75853030836057611687548079960, −6.91407771158084382717635796067, −6.07758903905233279709004495093, −5.54429293705556952252076270136, −3.69108900969403891272805861381, −2.68488419314965327664596288207, −1.03688728596379351834665441362,
1.81915205230920132428805659887, 2.30687747367505243841952125733, 4.03352010168491127725433684346, 5.17718559033199572169222629313, 6.14854343662923738938983136150, 7.15389532332101583042758512731, 8.165571058131080722760765304326, 8.955921144603274735581475111612, 9.290162366135699489565431496031, 10.41840607276034183894334153694