| L(s) = 1 | − 2.74·2-s − 3·3-s − 0.483·4-s + 19.4·5-s + 8.22·6-s + 7.48·7-s + 23.2·8-s + 9·9-s − 53.4·10-s + 22.8·11-s + 1.44·12-s − 13·13-s − 20.5·14-s − 58.4·15-s − 59.8·16-s + 67.0·17-s − 24.6·18-s + 16.5·19-s − 9.41·20-s − 22.4·21-s − 62.7·22-s − 175.·23-s − 69.7·24-s + 254.·25-s + 35.6·26-s − 27·27-s − 3.61·28-s + ⋯ |
| L(s) = 1 | − 0.969·2-s − 0.577·3-s − 0.0604·4-s + 1.74·5-s + 0.559·6-s + 0.404·7-s + 1.02·8-s + 0.333·9-s − 1.68·10-s + 0.627·11-s + 0.0348·12-s − 0.277·13-s − 0.391·14-s − 1.00·15-s − 0.935·16-s + 0.956·17-s − 0.323·18-s + 0.199·19-s − 0.105·20-s − 0.233·21-s − 0.608·22-s − 1.59·23-s − 0.593·24-s + 2.03·25-s + 0.268·26-s − 0.192·27-s − 0.0244·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8517795692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8517795692\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 2.74T + 8T^{2} \) |
| 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 - 7.48T + 343T^{2} \) |
| 11 | \( 1 - 22.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 67.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 251.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 79.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 194.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 734.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 933.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 557.T + 9.12e5T^{2} \) |
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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
−16.35389087788887356414339717339, −14.31455700784265084109245368927, −13.56200222549742787414274124646, −11.98693603499908101350498142670, −10.22365105567801067018029684852, −9.829271318127610106617054814351, −8.365770672875975183148628021105, −6.51744942665318506834786953848, −5.05952133174969234512209771076, −1.51620518894178504511611455851,
1.51620518894178504511611455851, 5.05952133174969234512209771076, 6.51744942665318506834786953848, 8.365770672875975183148628021105, 9.829271318127610106617054814351, 10.22365105567801067018029684852, 11.98693603499908101350498142670, 13.56200222549742787414274124646, 14.31455700784265084109245368927, 16.35389087788887356414339717339
