| L(s) = 1 | − 5·5-s + 7.95·7-s − 37.1·11-s − 35.1·13-s + 99.1·17-s + 44.6·19-s + 102.·23-s + 25·25-s − 285.·29-s − 238.·31-s − 39.7·35-s + 339.·37-s + 423.·41-s − 144.·43-s + 418.·47-s − 279.·49-s − 186.·53-s + 185.·55-s + 293.·59-s − 701.·61-s + 175.·65-s + 292.·67-s + 738.·71-s + 453.·73-s − 295.·77-s − 892.·79-s − 66.4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.429·7-s − 1.01·11-s − 0.749·13-s + 1.41·17-s + 0.539·19-s + 0.930·23-s + 0.200·25-s − 1.82·29-s − 1.38·31-s − 0.192·35-s + 1.50·37-s + 1.61·41-s − 0.511·43-s + 1.29·47-s − 0.815·49-s − 0.484·53-s + 0.455·55-s + 0.648·59-s − 1.47·61-s + 0.335·65-s + 0.533·67-s + 1.23·71-s + 0.726·73-s − 0.436·77-s − 1.27·79-s − 0.0878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 7.95T + 343T^{2} \) |
| 11 | \( 1 + 37.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 285.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 238.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 423.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 418.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 186.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 293.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 292.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 738.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 453.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 892.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 66.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 868.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 112.T + 9.12e5T^{2} \) |
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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
−8.000830147267542907370752733688, −7.69745882535506736497541107243, −7.06281835878687276647948262454, −5.63172908791594710264928167297, −5.31434426175996028334164554659, −4.27616309266689690879340741374, −3.30091148735065007498839838807, −2.43125884001732969422906838093, −1.17856952881819708577546137022, 0,
1.17856952881819708577546137022, 2.43125884001732969422906838093, 3.30091148735065007498839838807, 4.27616309266689690879340741374, 5.31434426175996028334164554659, 5.63172908791594710264928167297, 7.06281835878687276647948262454, 7.69745882535506736497541107243, 8.000830147267542907370752733688
