L(s) = 1 | + 4.46·5-s − 0.464·13-s + 7.92·17-s + 14.9·25-s + 8.46·29-s − 11.3·37-s − 10·41-s − 7·49-s + 14·53-s + 5.39·61-s − 2.07·65-s − 10.8·73-s + 35.3·85-s − 8.85·89-s + 18·97-s − 2·101-s + 20.3·109-s + 6.85·113-s + ⋯ |
L(s) = 1 | + 1.99·5-s − 0.128·13-s + 1.92·17-s + 2.98·25-s + 1.57·29-s − 1.87·37-s − 1.56·41-s − 49-s + 1.92·53-s + 0.690·61-s − 0.256·65-s − 1.27·73-s + 3.83·85-s − 0.938·89-s + 1.82·97-s − 0.199·101-s + 1.94·109-s + 0.644·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.320823343\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.320823343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.464T + 13T^{2} \) |
| 17 | \( 1 - 7.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 8.85T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.458559819197001230139919379589, −7.33803298218640757945740010859, −6.66644896368350423967254256432, −5.99481736709807693568510709748, −5.31611649341806400027763593090, −4.91146880242015936843557888247, −3.49055946466631169159982354762, −2.77035559384351876440051649161, −1.82638674527160399798748683099, −1.08041236900387801987144336625,
1.08041236900387801987144336625, 1.82638674527160399798748683099, 2.77035559384351876440051649161, 3.49055946466631169159982354762, 4.91146880242015936843557888247, 5.31611649341806400027763593090, 5.99481736709807693568510709748, 6.66644896368350423967254256432, 7.33803298218640757945740010859, 8.458559819197001230139919379589