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
\(L(1)\) |
\(\approx\) |
\(0.7122946214\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7122946214\) |
\(L(\frac{3}{2})\) |
|
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
−7.65415018819301913413088582857, −6.96042855913870144447067234567, −6.23244789216067646139826694000, −5.71728667293393440436333534588, −4.91543082608821415981115842113, −4.26552683599743235378258794440, −3.44866747889564632067088265608, −2.45725684525357114817815704027, −1.90887123902359963781301320386, −0.38093415458012687351822529968,
0.38093415458012687351822529968, 1.90887123902359963781301320386, 2.45725684525357114817815704027, 3.44866747889564632067088265608, 4.26552683599743235378258794440, 4.91543082608821415981115842113, 5.71728667293393440436333534588, 6.23244789216067646139826694000, 6.96042855913870144447067234567, 7.65415018819301913413088582857