L(s) = 1 | + (1.20 + 2.09i)3-s + (1.91 − 3.31i)5-s + (−1 + 2.44i)7-s + (−1.41 + 2.44i)9-s + (0.207 + 0.358i)11-s + 2.82·13-s + 9.24·15-s + (2.91 + 5.04i)17-s + (1.79 − 3.10i)19-s + (−6.32 + 0.866i)21-s + (−1.62 + 2.80i)23-s + (−4.82 − 8.36i)25-s + 0.414·27-s − 2.82·29-s + (−4.20 − 7.28i)31-s + ⋯ |
L(s) = 1 | + (0.696 + 1.20i)3-s + (0.856 − 1.48i)5-s + (−0.377 + 0.925i)7-s + (−0.471 + 0.816i)9-s + (0.0624 + 0.108i)11-s + 0.784·13-s + 2.38·15-s + (0.706 + 1.22i)17-s + (0.411 − 0.712i)19-s + (−1.38 + 0.188i)21-s + (−0.338 + 0.585i)23-s + (−0.965 − 1.67i)25-s + 0.0797·27-s − 0.525·29-s + (−0.755 − 1.30i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.84680 + 0.698283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84680 + 0.698283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 3 | \( 1 + (-1.20 - 2.09i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.91 + 3.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.207 - 0.358i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + (-2.91 - 5.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.79 + 3.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.62 - 2.80i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + (4.20 + 7.28i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.32 - 2.30i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + (3.79 - 6.56i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.44 + 7.70i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.32 - 2.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.62 + 9.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + (-1.67 - 2.89i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.03 + 6.98i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
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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
−11.03396320178398505286909849002, −9.809374890803133819076482821438, −9.449663538329648794725658824654, −8.779054673000626940984356941746, −8.061388243389131993724170010794, −6.07717756278121790203969837410, −5.42489252122897023509179551522, −4.39209296383255402598031449399, −3.31428296558592307270184770821, −1.75877139562812741930336576599,
1.47383197898904568937924673469, 2.77469537252977794273836367651, 3.57083554942833658290607770373, 5.65893516315521253914383452004, 6.71438932576127662253402213286, 7.12280170064331693340806830370, 7.963340141721173088402060331447, 9.232303939452808602406750101878, 10.19880947291778291110050070744, 10.81559682493514166558638141120