| L(s) = 1 | + (−1.93 − 1.93i)5-s − 1.41·7-s + (0.732 − 0.732i)11-s + (−1.73 − 1.73i)13-s + 5.27i·17-s + (−5.27 + 5.27i)19-s − 3.46i·23-s + 2.46i·25-s + (2.31 − 2.31i)29-s + 9.14i·31-s + (2.73 + 2.73i)35-s + (2.46 − 2.46i)37-s + 4.52·41-s + (3.48 + 3.48i)43-s + 10.3·47-s + ⋯ |
| L(s) = 1 | + (−0.863 − 0.863i)5-s − 0.534·7-s + (0.220 − 0.220i)11-s + (−0.480 − 0.480i)13-s + 1.28i·17-s + (−1.21 + 1.21i)19-s − 0.722i·23-s + 0.492i·25-s + (0.429 − 0.429i)29-s + 1.64i·31-s + (0.461 + 0.461i)35-s + (0.405 − 0.405i)37-s + 0.705·41-s + (0.531 + 0.531i)43-s + 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9238683389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9238683389\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.93 + 1.93i)T + 5iT^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + (-0.732 + 0.732i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.73 + 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 - 5.27iT - 17T^{2} \) |
| 19 | \( 1 + (5.27 - 5.27i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + (-2.31 + 2.31i)T - 29iT^{2} \) |
| 31 | \( 1 - 9.14iT - 31T^{2} \) |
| 37 | \( 1 + (-2.46 + 2.46i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 + (-3.48 - 3.48i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + (8.24 + 8.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (-8.92 + 8.92i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3 - 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.86 + 3.86i)T - 67iT^{2} \) |
| 71 | \( 1 - 14iT - 71T^{2} \) |
| 73 | \( 1 - 4.92iT - 73T^{2} \) |
| 79 | \( 1 + 2.17iT - 79T^{2} \) |
| 83 | \( 1 + (-10.7 - 10.7i)T + 83iT^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−8.863436912858532815623188546982, −8.303223350297269567585406557683, −7.84910702846788670939747140836, −6.66794666971632769077615367947, −6.05585800486433495740834010801, −5.04174125740675990487359273567, −4.14184754808213624288380257927, −3.59734722044519364415966080013, −2.27977215635795330248047735685, −0.890472788556210455252287597571,
0.41397637437365393184505697432, 2.32646648207678588739285485842, 3.02874855670634958802701592837, 4.06796322334976796697419518341, 4.71481974449848365639429902165, 5.93987526767845233564734750750, 6.82048668988711913067964088497, 7.24616383680800739837114462542, 7.945236401664261244446926305746, 9.183388142568971640413083972249
