| L(s) = 1 | + 243·3-s − 5.28e3·5-s − 2.49e4·7-s + 5.90e4·9-s − 2.27e5·11-s − 1.28e6·15-s − 6.06e6·21-s + 1.81e7·25-s + 1.43e7·27-s + 3.63e7·29-s + 8.95e6·31-s − 5.51e7·33-s + 1.31e8·35-s − 3.11e8·45-s + 3.40e8·49-s − 6.17e8·53-s + 1.19e9·55-s − 5.88e8·59-s − 1.47e9·63-s + 4.08e9·73-s + 4.40e9·75-s + 5.66e9·77-s + 5.86e9·79-s + 3.48e9·81-s − 7.87e9·83-s + 8.82e9·87-s + 2.17e9·93-s + ⋯ |
| L(s) = 1 | + 3-s − 1.69·5-s − 1.48·7-s + 9-s − 1.40·11-s − 1.69·15-s − 1.48·21-s + 1.85·25-s + 27-s + 1.77·29-s + 0.312·31-s − 1.40·33-s + 2.50·35-s − 1.69·45-s + 1.20·49-s − 1.47·53-s + 2.38·55-s − 0.823·59-s − 1.48·63-s + 1.96·73-s + 1.85·75-s + 2.09·77-s + 1.90·79-s + 81-s − 1.99·83-s + 1.77·87-s + 0.312·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.168613159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.168613159\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 + 5282 T + p^{10} T^{2} \) |
| 7 | \( 1 + 24950 T + p^{10} T^{2} \) |
| 11 | \( 1 + 227050 T + p^{10} T^{2} \) |
| 13 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 23 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 29 | \( 1 - 36304750 T + p^{10} T^{2} \) |
| 31 | \( 1 - 8955802 T + p^{10} T^{2} \) |
| 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 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 53 | \( 1 + 617985986 T + p^{10} T^{2} \) |
| 59 | \( 1 + 588563050 T + p^{10} T^{2} \) |
| 61 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 67 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( 1 - 4081435250 T + p^{10} T^{2} \) |
| 79 | \( 1 - 5863410298 T + p^{10} T^{2} \) |
| 83 | \( 1 + 7877173786 T + p^{10} T^{2} \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( 1 + 9031061950 T + p^{10} T^{2} \) |
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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
−12.27040797229239969080521096916, −10.74577890550363774737727686040, −9.735304985792071119913440767385, −8.441022660896353596607769369002, −7.70881284737300608400544406470, −6.66570295655420825858187479393, −4.62970825339288549100463800899, −3.43274519888888448201690433126, −2.76255533541464003257465615404, −0.52769180208082321220316023642,
0.52769180208082321220316023642, 2.76255533541464003257465615404, 3.43274519888888448201690433126, 4.62970825339288549100463800899, 6.66570295655420825858187479393, 7.70881284737300608400544406470, 8.441022660896353596607769369002, 9.735304985792071119913440767385, 10.74577890550363774737727686040, 12.27040797229239969080521096916
