L(s) = 1 | + 3.23·3-s + 1.23·5-s + 7-s + 7.47·9-s − 2.47·11-s − 5.23·13-s + 4.00·15-s − 4.47·17-s − 3.23·19-s + 3.23·21-s + 4·23-s − 3.47·25-s + 14.4·27-s − 4.47·29-s + 6.47·31-s − 8.00·33-s + 1.23·35-s − 4.47·37-s − 16.9·39-s + 0.472·41-s + 2.47·43-s + 9.23·45-s + 1.52·47-s + 49-s − 14.4·51-s + 10·53-s − 3.05·55-s + ⋯ |
L(s) = 1 | + 1.86·3-s + 0.552·5-s + 0.377·7-s + 2.49·9-s − 0.745·11-s − 1.45·13-s + 1.03·15-s − 1.08·17-s − 0.742·19-s + 0.706·21-s + 0.834·23-s − 0.694·25-s + 2.78·27-s − 0.830·29-s + 1.16·31-s − 1.39·33-s + 0.208·35-s − 0.735·37-s − 2.71·39-s + 0.0737·41-s + 0.376·43-s + 1.37·45-s + 0.222·47-s + 0.142·49-s − 2.02·51-s + 1.37·53-s − 0.412·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.602978127\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.602978127\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 0.472T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 + 6.76T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−10.78791531696252790432431091309, −9.928580108846962340176461681333, −9.241643254315026546593985324918, −8.425631430175080871150353911804, −7.62607127682092283007091123577, −6.80097529692576066096871324143, −5.10693491524925785420875481967, −4.10132448669145396627282680616, −2.65317914822623940275315184375, −2.07760455765410581888397145808,
2.07760455765410581888397145808, 2.65317914822623940275315184375, 4.10132448669145396627282680616, 5.10693491524925785420875481967, 6.80097529692576066096871324143, 7.62607127682092283007091123577, 8.425631430175080871150353911804, 9.241643254315026546593985324918, 9.928580108846962340176461681333, 10.78791531696252790432431091309