| L(s) = 1 | + 3.79e3·3-s − 8.06e6·5-s + 1.77e8·7-s − 1.14e9·9-s + 3.21e9·11-s + 1.43e10·13-s − 3.06e10·15-s + 4.05e11·17-s + 1.96e12·19-s + 6.72e11·21-s + 7.72e12·23-s + 4.59e13·25-s − 8.77e12·27-s − 5.61e13·29-s + 1.62e14·31-s + 1.22e13·33-s − 1.42e15·35-s + 3.64e14·37-s + 5.46e13·39-s − 1.20e15·41-s − 3.11e15·43-s + 9.25e15·45-s − 1.96e15·47-s + 1.99e16·49-s + 1.54e15·51-s + 3.16e15·53-s − 2.59e16·55-s + ⋯ |
| L(s) = 1 | + 0.111·3-s − 1.84·5-s + 1.65·7-s − 0.987·9-s + 0.411·11-s + 0.376·13-s − 0.205·15-s + 0.830·17-s + 1.39·19-s + 0.184·21-s + 0.894·23-s + 2.40·25-s − 0.221·27-s − 0.718·29-s + 1.10·31-s + 0.0458·33-s − 3.06·35-s + 0.461·37-s + 0.0419·39-s − 0.577·41-s − 0.943·43-s + 1.82·45-s − 0.255·47-s + 1.75·49-s + 0.0925·51-s + 0.131·53-s − 0.759·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(1.631964896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631964896\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 3.79e3T + 1.16e9T^{2} \) |
| 5 | \( 1 + 8.06e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.77e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 3.21e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 1.43e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.05e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.96e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 7.72e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 5.61e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.62e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 3.64e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.20e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 3.11e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.96e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.16e15T + 5.77e32T^{2} \) |
| 59 | \( 1 - 1.75e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 6.75e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 1.62e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 6.94e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 6.40e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.59e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 2.39e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 6.86e17T + 1.09e37T^{2} \) |
| 97 | \( 1 - 5.24e18T + 5.60e37T^{2} \) |
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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
−16.86037125627878683969062629733, −15.27998716978712455683903539730, −14.28385632090469457574961782005, −11.79951147661603960946027259868, −11.28850260427206759085098384652, −8.492543142689373971972239201083, −7.60625812446839756953236334818, −4.96225750193500310122462633565, −3.39932494505091093252316442104, −0.946380221864363290003340595567,
0.946380221864363290003340595567, 3.39932494505091093252316442104, 4.96225750193500310122462633565, 7.60625812446839756953236334818, 8.492543142689373971972239201083, 11.28850260427206759085098384652, 11.79951147661603960946027259868, 14.28385632090469457574961782005, 15.27998716978712455683903539730, 16.86037125627878683969062629733
