| L(s) = 1 | + 7·3-s + 16·4-s − 49·5-s − 32·9-s + 121·11-s + 112·12-s − 343·15-s + 256·16-s − 784·20-s + 167·23-s + 1.77e3·25-s − 791·27-s − 553·31-s + 847·33-s − 512·36-s − 2.11e3·37-s + 1.93e3·44-s + 1.56e3·45-s − 1.91e3·47-s + 1.79e3·48-s + 2.40e3·49-s − 718·53-s − 5.92e3·55-s + 4.48e3·59-s − 5.48e3·60-s + 4.09e3·64-s − 7.75e3·67-s + ⋯ |
| L(s) = 1 | + 7/9·3-s + 4-s − 1.95·5-s − 0.395·9-s + 11-s + 7/9·12-s − 1.52·15-s + 16-s − 1.95·20-s + 0.315·23-s + 2.84·25-s − 1.08·27-s − 0.575·31-s + 7/9·33-s − 0.395·36-s − 1.54·37-s + 44-s + 0.774·45-s − 0.868·47-s + 7/9·48-s + 49-s − 0.255·53-s − 1.95·55-s + 1.28·59-s − 1.52·60-s + 64-s − 1.72·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.206681487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206681487\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - p^{2} T \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 3 | \( 1 - 7 T + p^{4} T^{2} \) |
| 5 | \( 1 + 49 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 - 167 T + p^{4} T^{2} \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 553 T + p^{4} T^{2} \) |
| 37 | \( 1 + 2113 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 + 1918 T + p^{4} T^{2} \) |
| 53 | \( 1 + 718 T + p^{4} T^{2} \) |
| 59 | \( 1 - 4487 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 + 7753 T + p^{4} T^{2} \) |
| 71 | \( 1 - 7607 T + p^{4} T^{2} \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 + 6433 T + p^{4} T^{2} \) |
| 97 | \( 1 + 9793 T + p^{4} T^{2} \) |
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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
−19.75885494398119872252381855677, −19.21831500864514360034759155597, −16.68606130447771987237201821798, −15.47741591826257693856979377606, −14.58729990820295578316234063201, −12.14791799560031264064261886161, −11.18632149482410471750487909209, −8.524438658210435446274777614942, −7.18620362855703168198936251634, −3.49864671127385759292183735940,
3.49864671127385759292183735940, 7.18620362855703168198936251634, 8.524438658210435446274777614942, 11.18632149482410471750487909209, 12.14791799560031264064261886161, 14.58729990820295578316234063201, 15.47741591826257693856979377606, 16.68606130447771987237201821798, 19.21831500864514360034759155597, 19.75885494398119872252381855677
