| L(s) = 1 | − 2.64·2-s + 4.97·4-s − 4.34·7-s − 7.84·8-s − 2.53·11-s + 2.22·13-s + 11.4·14-s + 10.7·16-s − 4.51·17-s + 0.864·19-s + 6.69·22-s − 0.357·23-s − 5.88·26-s − 21.6·28-s + 4.40·29-s − 0.455·31-s − 12.7·32-s + 11.9·34-s + 2.37·37-s − 2.28·38-s − 2.77·41-s + 10.2·43-s − 12.5·44-s + 0.943·46-s − 7.92·47-s + 11.8·49-s + 11.0·52-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 2.48·4-s − 1.64·7-s − 2.77·8-s − 0.763·11-s + 0.617·13-s + 3.06·14-s + 2.69·16-s − 1.09·17-s + 0.198·19-s + 1.42·22-s − 0.0744·23-s − 1.15·26-s − 4.08·28-s + 0.818·29-s − 0.0817·31-s − 2.25·32-s + 2.04·34-s + 0.390·37-s − 0.370·38-s − 0.433·41-s + 1.56·43-s − 1.89·44-s + 0.139·46-s − 1.15·47-s + 1.69·49-s + 1.53·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2783759082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2783759082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 - 0.864T + 19T^{2} \) |
| 23 | \( 1 + 0.357T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 + 0.455T + 31T^{2} \) |
| 37 | \( 1 - 2.37T + 37T^{2} \) |
| 41 | \( 1 + 2.77T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 7.92T + 47T^{2} \) |
| 53 | \( 1 + 6.45T + 53T^{2} \) |
| 59 | \( 1 + 6.91T + 59T^{2} \) |
| 61 | \( 1 + 6.10T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 0.336T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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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
−8.257217462193477632707082355372, −7.57329463977554710558317947720, −6.84837921242582400893077338459, −6.33424894459716422567021164153, −5.76876146197788767543750120409, −4.36323498465833054556247520841, −3.09900661485572876339467889566, −2.72918901626822259869120433009, −1.57975170907842834972102518548, −0.36629999300746353425998270468,
0.36629999300746353425998270468, 1.57975170907842834972102518548, 2.72918901626822259869120433009, 3.09900661485572876339467889566, 4.36323498465833054556247520841, 5.76876146197788767543750120409, 6.33424894459716422567021164153, 6.84837921242582400893077338459, 7.57329463977554710558317947720, 8.257217462193477632707082355372
