| L(s) = 1 | − 0.568·2-s − 1.77·3-s − 1.67·4-s + 1.00·6-s − 2.89·7-s + 2.09·8-s + 0.142·9-s − 2.23·11-s + 2.97·12-s + 4.54·13-s + 1.64·14-s + 2.16·16-s + 1.59·17-s − 0.0807·18-s + 7.63·19-s + 5.12·21-s + 1.26·22-s + 5.02·23-s − 3.70·24-s − 2.58·26-s + 5.06·27-s + 4.84·28-s + 4.68·29-s − 0.922·31-s − 5.41·32-s + 3.95·33-s − 0.904·34-s + ⋯ |
| L(s) = 1 | − 0.401·2-s − 1.02·3-s − 0.838·4-s + 0.411·6-s − 1.09·7-s + 0.738·8-s + 0.0473·9-s − 0.672·11-s + 0.858·12-s + 1.26·13-s + 0.439·14-s + 0.541·16-s + 0.385·17-s − 0.0190·18-s + 1.75·19-s + 1.11·21-s + 0.270·22-s + 1.04·23-s − 0.756·24-s − 0.507·26-s + 0.974·27-s + 0.916·28-s + 0.870·29-s − 0.165·31-s − 0.956·32-s + 0.688·33-s − 0.155·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6065428547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6065428547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 0.568T + 2T^{2} \) |
| 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 - 7.63T + 19T^{2} \) |
| 23 | \( 1 - 5.02T + 23T^{2} \) |
| 29 | \( 1 - 4.68T + 29T^{2} \) |
| 31 | \( 1 + 0.922T + 31T^{2} \) |
| 37 | \( 1 + 4.28T + 37T^{2} \) |
| 41 | \( 1 + 4.67T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 2.34T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 5.97T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 8.70T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.049367575572086573272454302774, −7.44558341558561919625082234557, −6.48235199037068242389122587079, −5.98033463031838917048748150882, −5.16620873258646630484813976446, −4.77790538988997573095777061896, −3.40541572196320650667535097395, −3.16199236574532526292122016112, −1.31968902588238295816216239088, −0.52251960699901349393127798958,
0.52251960699901349393127798958, 1.31968902588238295816216239088, 3.16199236574532526292122016112, 3.40541572196320650667535097395, 4.77790538988997573095777061896, 5.16620873258646630484813976446, 5.98033463031838917048748150882, 6.48235199037068242389122587079, 7.44558341558561919625082234557, 8.049367575572086573272454302774
