| L(s) = 1 | − 39.6·2-s + 1.30e4·3-s − 1.29e5·4-s − 1.72e6·5-s − 5.16e5·6-s − 1.48e7·7-s + 1.03e7·8-s + 4.08e7·9-s + 6.83e7·10-s − 1.36e9·11-s − 1.68e9·12-s − 2.08e9·13-s + 5.87e8·14-s − 2.24e10·15-s + 1.65e10·16-s + 1.13e10·17-s − 1.61e9·18-s − 3.19e10·19-s + 2.23e11·20-s − 1.93e11·21-s + 5.40e10·22-s + 1.33e11·23-s + 1.34e11·24-s + 2.20e12·25-s + 8.27e10·26-s − 1.15e12·27-s + 1.91e12·28-s + ⋯ |
| L(s) = 1 | − 0.109·2-s + 1.14·3-s − 0.988·4-s − 1.97·5-s − 0.125·6-s − 0.971·7-s + 0.217·8-s + 0.316·9-s + 0.216·10-s − 1.91·11-s − 1.13·12-s − 0.709·13-s + 0.106·14-s − 2.26·15-s + 0.964·16-s + 0.394·17-s − 0.0346·18-s − 0.431·19-s + 1.94·20-s − 1.11·21-s + 0.209·22-s + 0.354·23-s + 0.249·24-s + 2.89·25-s + 0.0776·26-s − 0.784·27-s + 0.959·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.1610782603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1610782603\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 + 39.6T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.30e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 1.72e6T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.48e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.36e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.08e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.13e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 3.19e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.33e11T + 1.41e23T^{2} \) |
| 31 | \( 1 + 6.48e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.15e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 3.85e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 9.98e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.87e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 1.38e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 5.88e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.12e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.30e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.20e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 4.73e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.77e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.58e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.89e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.54e16T + 5.95e33T^{2} \) |
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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
−13.18962167674689142088437956892, −12.44045987443904222527794511389, −10.61048826894282167184770740797, −9.204068135287857834483723682703, −8.093725648045483647087227732435, −7.53536874873955172456392078912, −4.94117868856492206394209266387, −3.63235626270815702455873791533, −2.88726995251150686359478211275, −0.20118944854400433629959231473,
0.20118944854400433629959231473, 2.88726995251150686359478211275, 3.63235626270815702455873791533, 4.94117868856492206394209266387, 7.53536874873955172456392078912, 8.093725648045483647087227732435, 9.204068135287857834483723682703, 10.61048826894282167184770740797, 12.44045987443904222527794511389, 13.18962167674689142088437956892
