L(s) = 1 | + (−1.06 + 1.36i)3-s − 2.12i·5-s + i·7-s + (−0.732 − 2.90i)9-s + 5.81·11-s + 4.19·13-s + (2.90 + 2.26i)15-s + 5.81i·17-s + 2.73i·19-s + (−1.36 − 1.06i)21-s − 4.25·23-s + 0.464·25-s + (4.75 + 2.09i)27-s − 5.81i·29-s − 2.53i·31-s + ⋯ |
L(s) = 1 | + (−0.614 + 0.788i)3-s − 0.952i·5-s + 0.377i·7-s + (−0.244 − 0.969i)9-s + 1.75·11-s + 1.16·13-s + (0.751 + 0.585i)15-s + 1.41i·17-s + 0.626i·19-s + (−0.298 − 0.232i)21-s − 0.888·23-s + 0.0928·25-s + (0.914 + 0.403i)27-s − 1.08i·29-s − 0.455i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18100 + 0.230225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18100 + 0.230225i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.06 - 1.36i)T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 2.12iT - 5T^{2} \) |
| 11 | \( 1 - 5.81T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 - 5.81iT - 17T^{2} \) |
| 19 | \( 1 - 2.73iT - 19T^{2} \) |
| 23 | \( 1 + 4.25T + 23T^{2} \) |
| 29 | \( 1 + 5.81iT - 29T^{2} \) |
| 31 | \( 1 + 2.53iT - 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 1.55iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 1.55iT - 53T^{2} \) |
| 59 | \( 1 + 9.50T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−11.69724881393168504610600594115, −10.74619031227563891315961677829, −9.643842076198922595730161550664, −8.957484433441021290419974659563, −8.163936556075048517698774675164, −6.20242227917211859915451173756, −5.95712426372799149890226005004, −4.36071979960187832236372226593, −3.81520541167975223319159680453, −1.33268306698427048677751121155,
1.25528713615084735062295993055, 3.00856012859279898987969620565, 4.41615224709421167720096430448, 5.97326695759782600121999278361, 6.71842536928856475932545271068, 7.31492553754046380384057649296, 8.630219404306673968032721156825, 9.721400952860642380900626201625, 11.01055092506098433206032663908, 11.32832147425678561157120903778