| L(s) = 1 | + 1.41i·7-s + (−1 + i)13-s − 6·17-s + (−4.24 + 4.24i)19-s − 8.48i·23-s − 5i·25-s + (6 − 6i)29-s − 1.41·31-s + (−5 − 5i)37-s − 6i·41-s + (4.24 + 4.24i)43-s + 8.48·47-s + 5·49-s + (−6 − 6i)53-s + (−5 + 5i)61-s + ⋯ |
| L(s) = 1 | + 0.534i·7-s + (−0.277 + 0.277i)13-s − 1.45·17-s + (−0.973 + 0.973i)19-s − 1.76i·23-s − i·25-s + (1.11 − 1.11i)29-s − 0.254·31-s + (−0.821 − 0.821i)37-s − 0.937i·41-s + (0.646 + 0.646i)43-s + 1.23·47-s + 0.714·49-s + (−0.824 − 0.824i)53-s + (−0.640 + 0.640i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7645955147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7645955147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5iT^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (4.24 - 4.24i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 + (-6 + 6i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + (6 + 6i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (5 - 5i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.82 - 2.82i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 + (8.48 - 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.619654790358728473109093344444, −8.282426423453190551395694704965, −7.10426017755654082895921003572, −6.38600644488437615369102147000, −5.78839142806257455153022381807, −4.51067837790152683634900013540, −4.16797541630181040620123468212, −2.62008963743873140182542144088, −2.07623991382578500499876583397, −0.25745859414512768852423462568,
1.32019245594204322139546197352, 2.53127196562431032126093725849, 3.53393521309294082321785235390, 4.49445709084411454844533274266, 5.17469720094715453437008499386, 6.23288245415410835256997254398, 7.03908727371944970640168963106, 7.52258606871121673571445577925, 8.668648361362987387504220850957, 9.088366469215933333778243801089
