| L(s) = 1 | − 5.90e4·3-s − 2.12e7·5-s − 6.32e8·7-s + 3.48e9·9-s − 5.97e10·11-s + 7.38e11·13-s + 1.25e12·15-s − 8.35e12·17-s − 4.19e13·19-s + 3.73e13·21-s − 4.48e13·23-s − 2.41e13·25-s − 2.05e14·27-s − 2.76e15·29-s − 8.36e15·31-s + 3.52e15·33-s + 1.34e16·35-s − 1.77e16·37-s − 4.36e16·39-s + 1.45e17·41-s − 1.24e17·43-s − 7.41e16·45-s − 4.28e17·47-s − 1.59e17·49-s + 4.93e17·51-s − 4.77e17·53-s + 1.27e18·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.974·5-s − 0.845·7-s + 0.333·9-s − 0.694·11-s + 1.48·13-s + 0.562·15-s − 1.00·17-s − 1.56·19-s + 0.488·21-s − 0.225·23-s − 0.0505·25-s − 0.192·27-s − 1.22·29-s − 1.83·31-s + 0.401·33-s + 0.824·35-s − 0.607·37-s − 0.857·39-s + 1.69·41-s − 0.880·43-s − 0.324·45-s − 1.18·47-s − 0.284·49-s + 0.580·51-s − 0.374·53-s + 0.676·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.08481030669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08481030669\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.90e4T \) |
| good | 5 | \( 1 + 2.12e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 6.32e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 5.97e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.38e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 8.35e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.19e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 4.48e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.76e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 8.36e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.77e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.45e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.24e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.28e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 4.77e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.61e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 3.76e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.81e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.00e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.72e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 3.28e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 3.05e17T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.34e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 5.92e20T + 5.27e41T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.25106965839400563506751315848, −10.72259924372898671469954847623, −9.155447495857736348764367623546, −8.014248946255929123888854970691, −6.75565724177711607703954500965, −5.80802489782648943315384068513, −4.28088647509277283177005830284, −3.45474248086505151295524693937, −1.83900229815880311317218577738, −0.13082036356501804553445518374,
0.13082036356501804553445518374, 1.83900229815880311317218577738, 3.45474248086505151295524693937, 4.28088647509277283177005830284, 5.80802489782648943315384068513, 6.75565724177711607703954500965, 8.014248946255929123888854970691, 9.155447495857736348764367623546, 10.72259924372898671469954847623, 11.25106965839400563506751315848
