| L(s) = 1 | − 322.·3-s + 3.12e3·5-s − 4.63e4·7-s − 7.32e4·9-s + 3.17e5·11-s + 1.96e6·13-s − 1.00e6·15-s + 6.70e6·17-s + 3.15e6·19-s + 1.49e7·21-s + 2.51e7·23-s + 9.76e6·25-s + 8.07e7·27-s + 1.66e8·29-s − 2.25e8·31-s − 1.02e8·33-s − 1.44e8·35-s − 4.84e7·37-s − 6.34e8·39-s − 5.76e8·41-s + 1.91e8·43-s − 2.28e8·45-s + 5.49e8·47-s + 1.68e8·49-s − 2.16e9·51-s + 1.16e9·53-s + 9.93e8·55-s + ⋯ |
| L(s) = 1 | − 0.765·3-s + 0.447·5-s − 1.04·7-s − 0.413·9-s + 0.594·11-s + 1.47·13-s − 0.342·15-s + 1.14·17-s + 0.292·19-s + 0.797·21-s + 0.814·23-s + 0.199·25-s + 1.08·27-s + 1.50·29-s − 1.41·31-s − 0.455·33-s − 0.465·35-s − 0.114·37-s − 1.12·39-s − 0.776·41-s + 0.198·43-s − 0.184·45-s + 0.349·47-s + 0.0850·49-s − 0.876·51-s + 0.384·53-s + 0.266·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.387898365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387898365\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 3.12e3T \) |
| good | 3 | \( 1 + 322.T + 1.77e5T^{2} \) |
| 7 | \( 1 + 4.63e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.17e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.96e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.70e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 3.15e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.51e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.66e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.25e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.84e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.76e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.91e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 5.49e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.16e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 5.46e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.14e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.91e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.78e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.17e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.51e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.46e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.44e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.12e10T + 7.15e21T^{2} \) |
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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
−16.03458385884730938452451840670, −14.26538754392787211991185845183, −12.95910073158939843814038225275, −11.64173366638452906321561310187, −10.30189389876211047645969176013, −8.846390771794035425364759206308, −6.62050569531529265957826017842, −5.58859370301800126600593293055, −3.34581210425957557176286113218, −0.931129079255647287645324693411,
0.931129079255647287645324693411, 3.34581210425957557176286113218, 5.58859370301800126600593293055, 6.62050569531529265957826017842, 8.846390771794035425364759206308, 10.30189389876211047645969176013, 11.64173366638452906321561310187, 12.95910073158939843814038225275, 14.26538754392787211991185845183, 16.03458385884730938452451840670
