L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s − 7-s + 1.00i·9-s + (−0.707 − 0.707i)11-s − 13-s − 1.00·15-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)21-s + (−0.707 + 0.707i)23-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41·29-s + (−1 − i)31-s − 1.00i·33-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s − 7-s + 1.00i·9-s + (−0.707 − 0.707i)11-s − 13-s − 1.00·15-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)21-s + (−0.707 + 0.707i)23-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41·29-s + (−1 − i)31-s − 1.00i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4256964895\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4256964895\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 97 | \( 1 - iT - T^{2} \) |
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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
−9.443486573069889224623530458026, −8.338620451940871617257841409282, −7.952984303666684339889036427089, −7.19167151735463479224751410914, −6.30121317957980541918548519851, −5.41507255782921900074309712551, −4.48336305577612026022482814705, −3.38379500245223172845672267947, −3.26292253891980931021983842157, −2.18081387835582871067205734890,
0.21795921469723859976053278531, 1.74924256949693405064544122204, 2.86810454341017606897953671796, 3.46388173050373944225760552067, 4.60941118347262985626802340190, 5.27806807221718028352980916156, 6.47652212399546064399230112201, 7.16257066670000564285167798089, 7.64831481910651059926809831467, 8.456442070052667660688783528562