| L(s) = 1 | − 1.46i·3-s + 3.03i·5-s − 1.90i·7-s + 0.863·9-s + 1.80i·11-s + 1.35·13-s + 4.43·15-s + (3.82 + 1.54i)17-s − 8.15·19-s − 2.78·21-s − 4.71i·23-s − 4.22·25-s − 5.64i·27-s − 5.60i·29-s + 6.17i·31-s + ⋯ |
| L(s) = 1 | − 0.843i·3-s + 1.35i·5-s − 0.720i·7-s + 0.287·9-s + 0.543i·11-s + 0.375·13-s + 1.14·15-s + (0.927 + 0.374i)17-s − 1.87·19-s − 0.607·21-s − 0.983i·23-s − 0.844·25-s − 1.08i·27-s − 1.04i·29-s + 1.10i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.000586500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.000586500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-3.82 - 1.54i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.46iT - 3T^{2} \) |
| 5 | \( 1 - 3.03iT - 5T^{2} \) |
| 7 | \( 1 + 1.90iT - 7T^{2} \) |
| 11 | \( 1 - 1.80iT - 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 19 | \( 1 + 8.15T + 19T^{2} \) |
| 23 | \( 1 + 4.71iT - 23T^{2} \) |
| 29 | \( 1 + 5.60iT - 29T^{2} \) |
| 31 | \( 1 - 6.17iT - 31T^{2} \) |
| 37 | \( 1 + 5.86iT - 37T^{2} \) |
| 41 | \( 1 + 3.86iT - 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 7.47T + 53T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 3.12T + 67T^{2} \) |
| 71 | \( 1 - 6.11iT - 71T^{2} \) |
| 73 | \( 1 - 0.793iT - 73T^{2} \) |
| 79 | \( 1 + 5.43iT - 79T^{2} \) |
| 83 | \( 1 - 2.89T + 83T^{2} \) |
| 89 | \( 1 - 2.43T + 89T^{2} \) |
| 97 | \( 1 + 2.53iT - 97T^{2} \) |
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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
−8.200209532418424971622199122134, −7.30852404773639864597360956191, −7.14514491113526587172800020335, −6.36021967969173702807210000610, −5.80194338847980529333454090410, −4.27497740901733116267889142797, −3.93331157807447745817957422224, −2.64694574116745477393625895932, −2.04021375537868849431631741278, −0.790768897048833785025289000761,
0.865313614988407205843795392547, 1.93933983316202485437112147415, 3.20002901825339619330678133854, 4.10331702490947124006630917664, 4.65035703284479679661352208961, 5.50870449462115981638312853774, 5.90551033300333937447804694964, 7.06138055274728368852730471138, 8.093968793055759162300583667809, 8.574892367962886756799688929754
