| L(s) = 1 | + 361·3-s + 1.02e3·4-s − 3.12e3·5-s + 1.68e4·7-s + 7.12e4·9-s + 6.22e3·11-s + 3.69e5·12-s + 6.06e5·13-s − 1.12e6·15-s + 1.04e6·16-s − 2.62e6·17-s − 3.20e6·20-s + 6.06e6·21-s + 9.76e6·25-s + 4.41e6·27-s + 1.72e7·28-s + 3.66e7·29-s + 2.24e6·33-s − 5.25e7·35-s + 7.29e7·36-s + 2.18e8·39-s + 6.37e6·44-s − 2.22e8·45-s − 4.55e8·47-s + 3.78e8·48-s + 2.82e8·49-s − 9.45e8·51-s + ⋯ |
| L(s) = 1 | + 1.48·3-s + 4-s − 5-s + 7-s + 1.20·9-s + 0.0386·11-s + 1.48·12-s + 1.63·13-s − 1.48·15-s + 16-s − 1.84·17-s − 20-s + 1.48·21-s + 25-s + 0.307·27-s + 28-s + 1.78·29-s + 0.0574·33-s − 35-s + 1.20·36-s + 2.42·39-s + 0.0386·44-s − 1.20·45-s − 1.98·47-s + 1.48·48-s + 49-s − 2.74·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.892392514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.892392514\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{5} T \) |
| 7 | \( 1 - p^{5} T \) |
| good | 2 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 3 | \( 1 - 361 T + p^{10} T^{2} \) |
| 11 | \( 1 - 6227 T + p^{10} T^{2} \) |
| 13 | \( 1 - 606461 T + p^{10} T^{2} \) |
| 17 | \( 1 + 2620411 T + p^{10} T^{2} \) |
| 19 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 23 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 29 | \( 1 - 36611423 T + p^{10} T^{2} \) |
| 31 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 37 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 41 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 43 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 47 | \( 1 + 455938111 T + p^{10} T^{2} \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 67 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 71 | \( 1 - 253915202 T + p^{10} T^{2} \) |
| 73 | \( 1 + 3825881314 T + p^{10} T^{2} \) |
| 79 | \( 1 + 5204795077 T + p^{10} T^{2} \) |
| 83 | \( 1 - 3202399286 T + p^{10} T^{2} \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( 1 + 16228370611 T + p^{10} T^{2} \) |
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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
−14.49960121784296116425001632897, −13.28739505713110981042052235887, −11.64179967352822653077114126630, −10.78977908996535089474801819815, −8.649819401372595594241709072532, −8.099312946177827436209384067200, −6.74277823170058055265677186148, −4.21535592688483909191337345631, −2.89430006924687470992848460949, −1.51376304566424787315124050019,
1.51376304566424787315124050019, 2.89430006924687470992848460949, 4.21535592688483909191337345631, 6.74277823170058055265677186148, 8.099312946177827436209384067200, 8.649819401372595594241709072532, 10.78977908996535089474801819815, 11.64179967352822653077114126630, 13.28739505713110981042052235887, 14.49960121784296116425001632897
