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
−9.183388142568971640413083972249, −7.945236401664261244446926305746, −7.24616383680800739837114462542, −6.82048668988711913067964088497, −5.93987526767845233564734750750, −4.71481974449848365639429902165, −4.06796322334976796697419518341, −3.02874855670634958802701592837, −2.32646648207678588739285485842, −0.41397637437365393184505697432,
0.890472788556210455252287597571, 2.27977215635795330248047735685, 3.59734722044519364415966080013, 4.14184754808213624288380257927, 5.04174125740675990487359273567, 6.05585800486433495740834010801, 6.66794666971632769077615367947, 7.84910702846788670939747140836, 8.303223350297269567585406557683, 8.863436912858532815623188546982