L(s) = 1 | + 0.465·3-s − 2.78·9-s + 1.37·11-s + 5.12·13-s + 0.337·17-s − 5.99·19-s + 6.54·23-s − 2.69·27-s − 7.99·29-s + 7.24·31-s + 0.641·33-s + 6.05·37-s + 2.38·39-s − 6.68·41-s − 5.02·43-s − 6.64·47-s + 0.157·51-s + 4.87·53-s − 2.79·57-s − 0.602·59-s + 13.0·61-s + 12.1·67-s + 3.05·69-s − 1.39·71-s + 1.68·73-s + 7.61·79-s + 7.09·81-s + ⋯ |
L(s) = 1 | + 0.268·3-s − 0.927·9-s + 0.415·11-s + 1.42·13-s + 0.0819·17-s − 1.37·19-s + 1.36·23-s − 0.518·27-s − 1.48·29-s + 1.30·31-s + 0.111·33-s + 0.995·37-s + 0.381·39-s − 1.04·41-s − 0.766·43-s − 0.969·47-s + 0.0220·51-s + 0.669·53-s − 0.369·57-s − 0.0784·59-s + 1.67·61-s + 1.47·67-s + 0.367·69-s − 0.165·71-s + 0.197·73-s + 0.856·79-s + 0.788·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.167318783\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167318783\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.465T + 3T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 0.337T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 + 5.02T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 + 0.602T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 0.691T + 97T^{2} \) |
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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
−7.942858033581126263325368720716, −6.67336406055222354320448683435, −6.53005833698779580666007863440, −5.63844039275517370357363195491, −4.99260565234530339209282330194, −3.98526414359134779201418085931, −3.49290678244985203838124742775, −2.63967426829083068161470440240, −1.75616488834168473174567142423, −0.69433296574201664430344661263,
0.69433296574201664430344661263, 1.75616488834168473174567142423, 2.63967426829083068161470440240, 3.49290678244985203838124742775, 3.98526414359134779201418085931, 4.99260565234530339209282330194, 5.63844039275517370357363195491, 6.53005833698779580666007863440, 6.67336406055222354320448683435, 7.942858033581126263325368720716