| L(s) = 1 | − 3.25·3-s + 3.01·5-s + 1.05·7-s + 7.62·9-s + 0.331·11-s + 5.94·13-s − 9.83·15-s − 5.88·17-s − 19-s − 3.43·21-s − 7.02·23-s + 4.11·25-s − 15.0·27-s − 0.630·29-s + 1.75·31-s − 1.08·33-s + 3.18·35-s − 8.41·37-s − 19.3·39-s − 3.43·41-s − 12.5·43-s + 23.0·45-s − 11.6·47-s − 5.88·49-s + 19.1·51-s + 53-s + 1.00·55-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 1.35·5-s + 0.398·7-s + 2.54·9-s + 0.0999·11-s + 1.65·13-s − 2.54·15-s − 1.42·17-s − 0.229·19-s − 0.750·21-s − 1.46·23-s + 0.822·25-s − 2.90·27-s − 0.117·29-s + 0.314·31-s − 0.188·33-s + 0.538·35-s − 1.38·37-s − 3.10·39-s − 0.535·41-s − 1.90·43-s + 3.43·45-s − 1.70·47-s − 0.841·49-s + 2.68·51-s + 0.137·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 - 3.01T + 5T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 0.331T + 11T^{2} \) |
| 13 | \( 1 - 5.94T + 13T^{2} \) |
| 17 | \( 1 + 5.88T + 17T^{2} \) |
| 23 | \( 1 + 7.02T + 23T^{2} \) |
| 29 | \( 1 + 0.630T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 59 | \( 1 - 5.04T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 - 3.45T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 1.29T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 0.759T + 89T^{2} \) |
| 97 | \( 1 + 7.99T + 97T^{2} \) |
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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
−8.138956341732645480190678003927, −6.68544450547090572804448538708, −6.58071044107221361578519154906, −5.89377999298108925965612818424, −5.26869370937162429175098214545, −4.58025819682148579971883761272, −3.67806271587458605571926056262, −1.89447757224690494296071058385, −1.49039218470837446447822748832, 0,
1.49039218470837446447822748832, 1.89447757224690494296071058385, 3.67806271587458605571926056262, 4.58025819682148579971883761272, 5.26869370937162429175098214545, 5.89377999298108925965612818424, 6.58071044107221361578519154906, 6.68544450547090572804448538708, 8.138956341732645480190678003927
