L(s) = 1 | + (1 − i)2-s − 2i·4-s + 3.46·5-s + (−1.73 − 2i)7-s + (−2 − 2i)8-s + (3.46 − 3.46i)10-s − 2·11-s + 3.46·13-s + (−3.73 − 0.267i)14-s − 4·16-s − 3.46i·17-s + 6.92i·19-s − 6.92i·20-s + (−2 + 2i)22-s + 2i·23-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s + 1.54·5-s + (−0.654 − 0.755i)7-s + (−0.707 − 0.707i)8-s + (1.09 − 1.09i)10-s − 0.603·11-s + 0.960·13-s + (−0.997 − 0.0716i)14-s − 16-s − 0.840i·17-s + 1.58i·19-s − 1.54i·20-s + (−0.426 + 0.426i)22-s + 0.417i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0716 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0716 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61634 - 1.73655i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61634 - 1.73655i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 6.92iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 - 6.92iT - 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−10.59507753691665026707017781771, −9.820300247458261435513703606238, −9.571254280974355735709968044641, −8.013640430106141172242941288110, −6.46536442656294721357299602826, −6.03518881728731297856626236946, −4.99909179566428918993886433152, −3.71385986925118541748507299125, −2.59893755589607865092552444958, −1.29168738271627690120488975094,
2.22841257062652482863833157614, 3.22461266833243799297888382303, 4.81565402844349154016123286053, 5.76889489847241323040371488662, 6.23466015417750218056857508454, 7.18973410787547233367326601808, 8.763285279184989153050754896489, 8.976649843788313505550426384224, 10.26071603594475396620348973877, 11.12789746312879149087340185997