| L(s) = 1 | + 4.80e5·2-s + 9.33e10·4-s − 1.38e13·5-s − 5.42e15·7-s − 2.11e16·8-s − 6.64e18·10-s + 1.03e18·11-s − 1.23e19·13-s − 2.60e21·14-s − 2.30e22·16-s + 5.41e22·17-s − 3.08e23·19-s − 1.29e24·20-s + 4.96e23·22-s + 2.61e25·23-s + 1.18e26·25-s − 5.92e24·26-s − 5.06e26·28-s − 2.49e26·29-s − 2.34e27·31-s − 8.14e27·32-s + 2.60e28·34-s + 7.49e28·35-s + 6.21e28·37-s − 1.48e29·38-s + 2.92e29·40-s + 8.53e29·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.679·4-s − 1.62·5-s − 1.25·7-s − 0.415·8-s − 2.10·10-s + 0.0560·11-s − 0.0303·13-s − 1.63·14-s − 1.21·16-s + 0.934·17-s − 0.680·19-s − 1.10·20-s + 0.0725·22-s + 1.68·23-s + 1.62·25-s − 0.0393·26-s − 0.854·28-s − 0.219·29-s − 0.602·31-s − 1.16·32-s + 1.21·34-s + 2.04·35-s + 0.605·37-s − 0.881·38-s + 0.673·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(\approx\) |
\(1.659455546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659455546\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 4.80e5T + 1.37e11T^{2} \) |
| 5 | \( 1 + 1.38e13T + 7.27e25T^{2} \) |
| 7 | \( 1 + 5.42e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 1.03e18T + 3.40e38T^{2} \) |
| 13 | \( 1 + 1.23e19T + 1.64e41T^{2} \) |
| 17 | \( 1 - 5.41e22T + 3.36e45T^{2} \) |
| 19 | \( 1 + 3.08e23T + 2.06e47T^{2} \) |
| 23 | \( 1 - 2.61e25T + 2.42e50T^{2} \) |
| 29 | \( 1 + 2.49e26T + 1.28e54T^{2} \) |
| 31 | \( 1 + 2.34e27T + 1.51e55T^{2} \) |
| 37 | \( 1 - 6.21e28T + 1.05e58T^{2} \) |
| 41 | \( 1 - 8.53e29T + 4.70e59T^{2} \) |
| 43 | \( 1 - 3.53e29T + 2.74e60T^{2} \) |
| 47 | \( 1 + 6.68e29T + 7.37e61T^{2} \) |
| 53 | \( 1 + 1.24e32T + 6.28e63T^{2} \) |
| 59 | \( 1 + 6.57e31T + 3.32e65T^{2} \) |
| 61 | \( 1 + 1.06e33T + 1.14e66T^{2} \) |
| 67 | \( 1 + 8.44e33T + 3.67e67T^{2} \) |
| 71 | \( 1 - 7.04e33T + 3.13e68T^{2} \) |
| 73 | \( 1 - 9.36e33T + 8.76e68T^{2} \) |
| 79 | \( 1 - 1.51e35T + 1.63e70T^{2} \) |
| 83 | \( 1 - 1.43e35T + 1.01e71T^{2} \) |
| 89 | \( 1 + 2.28e35T + 1.34e72T^{2} \) |
| 97 | \( 1 - 7.07e36T + 3.24e73T^{2} \) |
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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
−12.97924981677093867485116166361, −12.32977298346883012399583422873, −11.08638761826484467966094767741, −9.141835712938481320721142610050, −7.47818031133602201288462997171, −6.23557045964603883301190964202, −4.72121089856995631565104750498, −3.63356016971806899270931581536, −2.98838845480565957989625613917, −0.52000855062034144903580596604,
0.52000855062034144903580596604, 2.98838845480565957989625613917, 3.63356016971806899270931581536, 4.72121089856995631565104750498, 6.23557045964603883301190964202, 7.47818031133602201288462997171, 9.141835712938481320721142610050, 11.08638761826484467966094767741, 12.32977298346883012399583422873, 12.97924981677093867485116166361
