| L(s) = 1 | − 9·3-s − 25·5-s + 81·9-s − 238·11-s − 142·13-s + 225·15-s − 98·17-s − 862·23-s + 625·25-s − 729·27-s + 238·29-s − 1.44e3·31-s + 2.14e3·33-s − 1.58e3·37-s + 1.27e3·39-s − 1.77e3·43-s − 2.02e3·45-s − 3.26e3·47-s + 2.40e3·49-s + 882·51-s + 5.95e3·55-s + 2.64e3·59-s + 3.55e3·65-s + 8.30e3·67-s + 7.75e3·69-s − 5.62e3·75-s + 1.10e4·79-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 9-s − 1.96·11-s − 0.840·13-s + 15-s − 0.339·17-s − 1.62·23-s + 25-s − 27-s + 0.282·29-s − 1.50·31-s + 1.96·33-s − 1.15·37-s + 0.840·39-s − 0.961·43-s − 45-s − 1.47·47-s + 49-s + 0.339·51-s + 1.96·55-s + 0.758·59-s + 0.840·65-s + 1.84·67-s + 1.62·69-s − 75-s + 1.76·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1969021365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1969021365\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 + 238 T + p^{4} T^{2} \) |
| 13 | \( 1 + 142 T + p^{4} T^{2} \) |
| 17 | \( 1 + 98 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 + 862 T + p^{4} T^{2} \) |
| 29 | \( 1 - 238 T + p^{4} T^{2} \) |
| 31 | \( 1 + 1442 T + p^{4} T^{2} \) |
| 37 | \( 1 + 1582 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 + 1778 T + p^{4} T^{2} \) |
| 47 | \( 1 + 3262 T + p^{4} T^{2} \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 - 2642 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 - 8302 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 - 11038 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−10.51951303167860309326881351383, −9.818160761937493369450732780375, −8.301002579850423812928226235395, −7.61228967662521854023272892428, −6.79655617124408819614961142362, −5.45318762300747542640050387652, −4.85290205837521688189641646844, −3.67233927446548864966275713700, −2.17058332802679564052012913008, −0.24412935431758469478699282036,
0.24412935431758469478699282036, 2.17058332802679564052012913008, 3.67233927446548864966275713700, 4.85290205837521688189641646844, 5.45318762300747542640050387652, 6.79655617124408819614961142362, 7.61228967662521854023272892428, 8.301002579850423812928226235395, 9.818160761937493369450732780375, 10.51951303167860309326881351383
