| L(s) = 1 | − i·3-s + (−2.06 + 0.857i)5-s − 3.22i·7-s − 9-s + 0.0657·11-s − 6.97i·13-s + (0.857 + 2.06i)15-s + 5.91i·17-s − 8.45·19-s − 3.22·21-s − i·23-s + (3.53 − 3.54i)25-s + i·27-s − 1.27·29-s + 6.03·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.923 + 0.383i)5-s − 1.22i·7-s − 0.333·9-s + 0.0198·11-s − 1.93i·13-s + (0.221 + 0.533i)15-s + 1.43i·17-s − 1.93·19-s − 0.704·21-s − 0.208i·23-s + (0.706 − 0.708i)25-s + 0.192i·27-s − 0.236·29-s + 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.004450180843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004450180843\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2.06 - 0.857i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 + 3.22iT - 7T^{2} \) |
| 11 | \( 1 - 0.0657T + 11T^{2} \) |
| 13 | \( 1 + 6.97iT - 13T^{2} \) |
| 17 | \( 1 - 5.91iT - 17T^{2} \) |
| 19 | \( 1 + 8.45T + 19T^{2} \) |
| 29 | \( 1 + 1.27T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 - 7.79iT - 37T^{2} \) |
| 41 | \( 1 + 9.18T + 41T^{2} \) |
| 43 | \( 1 + 6.44iT - 43T^{2} \) |
| 47 | \( 1 + 8.47iT - 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 0.969T + 59T^{2} \) |
| 61 | \( 1 - 4.13T + 61T^{2} \) |
| 67 | \( 1 + 1.43iT - 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 + 0.325iT - 73T^{2} \) |
| 79 | \( 1 + 0.193T + 79T^{2} \) |
| 83 | \( 1 - 7.81iT - 83T^{2} \) |
| 89 | \( 1 - 7.43T + 89T^{2} \) |
| 97 | \( 1 - 13.9iT - 97T^{2} \) |
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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
−8.580886644342733529084980528921, −8.185713450007012324255353776653, −7.68625855513525236489434527469, −6.73136421088095623162142023646, −6.32330118832582844935611293088, −5.14182965900276056246966715165, −4.08959732189956702496400089271, −3.55433712438353646227838057568, −2.49716662075904438725984125185, −1.07481629199820397212503009429,
0.00160260589061906418665963173, 1.90656884964015459887069518228, 2.82834312854728119779711252500, 4.00032099205846321251303914361, 4.56544154425680143651977465138, 5.25979819264691891052000655457, 6.36613468587809203446277225706, 6.95225675815856191357648489122, 8.068854012011773471354483253216, 8.732943085843753157042433785950
