L(s) = 1 | − 3-s + 0.344·5-s + 2.72·7-s + 9-s − 0.712·11-s + 6.42·13-s − 0.344·15-s − 5.96·17-s + 1.12·19-s − 2.72·21-s + 2.47·23-s − 4.88·25-s − 27-s + 1.35·29-s + 3.54·31-s + 0.712·33-s + 0.940·35-s + 8.77·37-s − 6.42·39-s + 3.02·41-s + 12.8·43-s + 0.344·45-s − 9.01·47-s + 0.450·49-s + 5.96·51-s − 5.26·53-s − 0.245·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.154·5-s + 1.03·7-s + 0.333·9-s − 0.214·11-s + 1.78·13-s − 0.0889·15-s − 1.44·17-s + 0.257·19-s − 0.595·21-s + 0.517·23-s − 0.976·25-s − 0.192·27-s + 0.252·29-s + 0.636·31-s + 0.124·33-s + 0.158·35-s + 1.44·37-s − 1.02·39-s + 0.472·41-s + 1.95·43-s + 0.0513·45-s − 1.31·47-s + 0.0642·49-s + 0.835·51-s − 0.723·53-s − 0.0331·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.080270396\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.080270396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
good | 5 | \( 1 - 0.344T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 + 0.712T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 - 8.77T + 37T^{2} \) |
| 41 | \( 1 - 3.02T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 + 9.01T + 47T^{2} \) |
| 53 | \( 1 + 5.26T + 53T^{2} \) |
| 59 | \( 1 - 9.02T + 59T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 4.01T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 + 1.26T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 0.273T + 89T^{2} \) |
| 97 | \( 1 + 0.105T + 97T^{2} \) |
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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
−8.092142697443849080566015549357, −7.42394459928526970590703059308, −6.41113018670872298591712217088, −6.07138076519907166166247601043, −5.20950263490056695915742054286, −4.45291605040224917296248314382, −3.89803802320985199495265695922, −2.66298586481151517461376066207, −1.69831068641184706248521263847, −0.828195383406123268083611027347,
0.828195383406123268083611027347, 1.69831068641184706248521263847, 2.66298586481151517461376066207, 3.89803802320985199495265695922, 4.45291605040224917296248314382, 5.20950263490056695915742054286, 6.07138076519907166166247601043, 6.41113018670872298591712217088, 7.42394459928526970590703059308, 8.092142697443849080566015549357