| L(s) = 1 | − 1.37·2-s − 0.109·4-s − 0.874·5-s − 1.13·7-s + 2.90·8-s + 1.20·10-s + 0.0912·11-s − 6.53·13-s + 1.56·14-s − 3.76·16-s − 3.65·17-s − 19-s + 0.0960·20-s − 0.125·22-s + 5·23-s − 4.23·25-s + 8.98·26-s + 0.125·28-s − 2.53·29-s − 7.09·31-s − 0.620·32-s + 5.01·34-s + 0.996·35-s − 0.716·37-s + 1.37·38-s − 2.53·40-s + 3.80·41-s + ⋯ |
| L(s) = 1 | − 0.972·2-s − 0.0549·4-s − 0.391·5-s − 0.430·7-s + 1.02·8-s + 0.380·10-s + 0.0275·11-s − 1.81·13-s + 0.418·14-s − 0.942·16-s − 0.885·17-s − 0.229·19-s + 0.0214·20-s − 0.0267·22-s + 1.04·23-s − 0.847·25-s + 1.76·26-s + 0.0236·28-s − 0.470·29-s − 1.27·31-s − 0.109·32-s + 0.860·34-s + 0.168·35-s − 0.117·37-s + 0.223·38-s − 0.401·40-s + 0.594·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1751792608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1751792608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 5 | \( 1 + 0.874T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 - 0.0912T + 11T^{2} \) |
| 13 | \( 1 + 6.53T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 7.09T + 31T^{2} \) |
| 37 | \( 1 + 0.716T + 37T^{2} \) |
| 41 | \( 1 - 3.80T + 41T^{2} \) |
| 43 | \( 1 + 0.158T + 43T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 + 6.90T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 - 7.76T + 89T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
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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
−7.63270272827856459091774686736, −7.46865924063907415261101238123, −6.77887681297941833782413598742, −5.79535501986866847204715905595, −4.85382647286540614117526552695, −4.43721729183695193739065572488, −3.46575142894546699323168025851, −2.47731878097755983076082686958, −1.63660673653816542591860589761, −0.23645877233279182102764693872,
0.23645877233279182102764693872, 1.63660673653816542591860589761, 2.47731878097755983076082686958, 3.46575142894546699323168025851, 4.43721729183695193739065572488, 4.85382647286540614117526552695, 5.79535501986866847204715905595, 6.77887681297941833782413598742, 7.46865924063907415261101238123, 7.63270272827856459091774686736
