| L(s) = 1 | + 2.07·2-s + 2.31·4-s + 1.54·5-s + 2.97·7-s + 0.657·8-s + 3.21·10-s + 11-s + 0.922·13-s + 6.17·14-s − 3.26·16-s + 4.29·17-s + 19-s + 3.58·20-s + 2.07·22-s − 1.21·23-s − 2.60·25-s + 1.91·26-s + 6.89·28-s − 3.55·29-s + 1.76·31-s − 8.10·32-s + 8.91·34-s + 4.60·35-s + 5.79·37-s + 2.07·38-s + 1.01·40-s − 2.84·41-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 1.15·4-s + 0.691·5-s + 1.12·7-s + 0.232·8-s + 1.01·10-s + 0.301·11-s + 0.255·13-s + 1.65·14-s − 0.816·16-s + 1.04·17-s + 0.229·19-s + 0.801·20-s + 0.442·22-s − 0.253·23-s − 0.521·25-s + 0.375·26-s + 1.30·28-s − 0.660·29-s + 0.316·31-s − 1.43·32-s + 1.52·34-s + 0.777·35-s + 0.953·37-s + 0.337·38-s + 0.160·40-s − 0.444·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.959214093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.959214093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 13 | \( 1 - 0.922T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 23 | \( 1 + 1.21T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 0.365T + 47T^{2} \) |
| 53 | \( 1 + 1.24T + 53T^{2} \) |
| 59 | \( 1 + 9.56T + 59T^{2} \) |
| 61 | \( 1 - 5.32T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 - 5.88T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 8.59T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.319364390278910711465940777031, −8.294426415145011299661293009247, −7.53346376819584524569801668856, −6.49415547563276018031816101044, −5.73412596389963614058224379453, −5.22852929984384696291720256752, −4.35404680575903596926805556473, −3.55058420766977864399344078983, −2.44683697479227034316458098673, −1.44928794180473953242019475378,
1.44928794180473953242019475378, 2.44683697479227034316458098673, 3.55058420766977864399344078983, 4.35404680575903596926805556473, 5.22852929984384696291720256752, 5.73412596389963614058224379453, 6.49415547563276018031816101044, 7.53346376819584524569801668856, 8.294426415145011299661293009247, 9.319364390278910711465940777031
