| L(s) = 1 | + 2.42i·3-s + 0.343i·5-s + (−2.23 − 1.42i)7-s − 2.86·9-s + (−2.51 − 2.16i)11-s − 13-s − 0.831·15-s + 7.55·17-s − 4.71·19-s + (3.44 − 5.40i)21-s + 7.48·23-s + 4.88·25-s + 0.324i·27-s − 4.61i·29-s + 6.22i·31-s + ⋯ |
| L(s) = 1 | + 1.39i·3-s + 0.153i·5-s + (−0.843 − 0.537i)7-s − 0.955·9-s + (−0.757 − 0.653i)11-s − 0.277·13-s − 0.214·15-s + 1.83·17-s − 1.08·19-s + (0.751 − 1.17i)21-s + 1.56·23-s + 0.976·25-s + 0.0624i·27-s − 0.857i·29-s + 1.11i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450096290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450096290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.42i)T \) |
| 11 | \( 1 + (2.51 + 2.16i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.42iT - 3T^{2} \) |
| 5 | \( 1 - 0.343iT - 5T^{2} \) |
| 17 | \( 1 - 7.55T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 + 4.61iT - 29T^{2} \) |
| 31 | \( 1 - 6.22iT - 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 - 7.26T + 41T^{2} \) |
| 43 | \( 1 - 1.07iT - 43T^{2} \) |
| 47 | \( 1 + 1.00iT - 47T^{2} \) |
| 53 | \( 1 + 1.66T + 53T^{2} \) |
| 59 | \( 1 + 0.229iT - 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 0.898T + 67T^{2} \) |
| 71 | \( 1 + 7.01T + 71T^{2} \) |
| 73 | \( 1 + 5.83T + 73T^{2} \) |
| 79 | \( 1 - 5.70iT - 79T^{2} \) |
| 83 | \( 1 - 5.78T + 83T^{2} \) |
| 89 | \( 1 + 0.531iT - 89T^{2} \) |
| 97 | \( 1 - 9.91iT - 97T^{2} \) |
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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
−8.899114339165613751957461170668, −8.011759469974429087731616923922, −7.20532039762045629557575059465, −6.39347465872564521826925637176, −5.46907908637279622095256873815, −4.93919433472225330321062444453, −4.01689483471447361349032468143, −3.26625013359047479287093071781, −2.79479681733454729322772060036, −0.899816104262319277839666463144,
0.54025744998178640199891163219, 1.63878403569426106260112088319, 2.61778858580072405926142070358, 3.20790152403196096773466509069, 4.57902638087600001942059325967, 5.49542117629738100492659478641, 6.05267829014028940679003785712, 7.06029714794931384326802740291, 7.23896584757445247057265064188, 8.134399842095558366572286050892
