| L(s) = 1 | + 0.792i·3-s + (−0.686 + 2.12i)5-s − 3.37·7-s + 2.37·9-s + 5.04i·11-s + (1 + 3.46i)13-s + (−1.68 − 0.543i)15-s − 2.67i·17-s − 3.46i·19-s − 2.67i·21-s − 6.63i·23-s + (−4.05 − 2.92i)25-s + 4.25i·27-s − 8.74·29-s + 3.46i·31-s + ⋯ |
| L(s) = 1 | + 0.457i·3-s + (−0.306 + 0.951i)5-s − 1.27·7-s + 0.790·9-s + 1.52i·11-s + (0.277 + 0.960i)13-s + (−0.435 − 0.140i)15-s − 0.648i·17-s − 0.794i·19-s − 0.583i·21-s − 1.38i·23-s + (−0.811 − 0.584i)25-s + 0.819i·27-s − 1.62·29-s + 0.622i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7519872956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7519872956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.686 - 2.12i)T \) |
| 13 | \( 1 + (-1 - 3.46i)T \) |
| good | 3 | \( 1 - 0.792iT - 3T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 5.04iT - 11T^{2} \) |
| 17 | \( 1 + 2.67iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6.63iT - 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 - 3.16iT - 41T^{2} \) |
| 43 | \( 1 - 9.30iT - 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 + 1.58iT - 53T^{2} \) |
| 59 | \( 1 + 6.63iT - 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 3.96iT - 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 + 8.51iT - 89T^{2} \) |
| 97 | \( 1 - 4.74T + 97T^{2} \) |
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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
−10.13154151641696887496703950941, −9.686470109719633094470953203419, −9.005114682161012434473373094021, −7.54365491485292367246479536392, −6.80526835972728989592724140279, −6.55624812140171852299232309750, −4.91730791073943375296738165090, −4.12570162011379962337667442254, −3.21494208131970797880474949107, −2.06532325746678494700357067828,
0.33514818386342813165710894177, 1.60934760200905005700198667783, 3.44160881432698790349330473059, 3.82547316460620827519997898517, 5.55808146181171028264368712793, 5.86986576893026860381060450440, 7.07827156611618828488171004361, 7.900139579518089598155795108438, 8.632877679375457245220074777844, 9.477523940094648296086002486234
