| L(s) = 1 | + 512·2-s + 2.62e5·4-s − 1.95e6·5-s + 2.66e7·7-s + 1.34e8·8-s − 1.00e9·10-s + 8.16e9·11-s + 4.92e10·13-s + 1.36e10·14-s + 6.87e10·16-s + 4.41e11·17-s + 1.79e12·19-s − 5.12e11·20-s + 4.18e12·22-s − 6.42e12·23-s + 3.81e12·25-s + 2.52e13·26-s + 6.99e12·28-s − 5.26e13·29-s − 5.69e12·31-s + 3.51e13·32-s + 2.25e14·34-s − 5.21e13·35-s − 1.22e15·37-s + 9.20e14·38-s − 2.62e14·40-s − 7.91e14·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.249·7-s + 0.353·8-s − 0.316·10-s + 1.04·11-s + 1.28·13-s + 0.176·14-s + 0.250·16-s + 0.902·17-s + 1.27·19-s − 0.223·20-s + 0.738·22-s − 0.743·23-s + 0.199·25-s + 0.910·26-s + 0.124·28-s − 0.673·29-s − 0.0386·31-s + 0.176·32-s + 0.638·34-s − 0.111·35-s − 1.55·37-s + 0.904·38-s − 0.158·40-s − 0.377·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(4.665629021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.665629021\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 512T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 1.95e6T \) |
| good | 7 | \( 1 - 2.66e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 8.16e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.92e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.41e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.79e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 6.42e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 5.26e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 5.69e12T + 2.16e28T^{2} \) |
| 37 | \( 1 + 1.22e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 7.91e14T + 4.39e30T^{2} \) |
| 43 | \( 1 - 1.58e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 1.36e16T + 5.88e31T^{2} \) |
| 53 | \( 1 + 4.14e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 9.46e15T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.17e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 1.04e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.14e15T + 1.49e35T^{2} \) |
| 73 | \( 1 - 9.22e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 9.89e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.15e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 7.12e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 2.58e17T + 5.60e37T^{2} \) |
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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
−10.90601698269867196467125097684, −9.568059860341260300756109449212, −8.346880498884958636433307034957, −7.30198817226727523076183483240, −6.18585813081540811119188588096, −5.20796007333053588260582786027, −3.87714977606251571154103017501, −3.36578657077033485212336318071, −1.74005838281878425021096365571, −0.864680515707330279566569761592,
0.864680515707330279566569761592, 1.74005838281878425021096365571, 3.36578657077033485212336318071, 3.87714977606251571154103017501, 5.20796007333053588260582786027, 6.18585813081540811119188588096, 7.30198817226727523076183483240, 8.346880498884958636433307034957, 9.568059860341260300756109449212, 10.90601698269867196467125097684
