| L(s) = 1 | − 2-s − 1.88·3-s + 4-s + 4.44·5-s + 1.88·6-s − 1.18·7-s − 8-s + 0.558·9-s − 4.44·10-s − 3.77·11-s − 1.88·12-s + 0.131·13-s + 1.18·14-s − 8.38·15-s + 16-s + 4.90·17-s − 0.558·18-s + 4.40·19-s + 4.44·20-s + 2.23·21-s + 3.77·22-s + 0.903·23-s + 1.88·24-s + 14.7·25-s − 0.131·26-s + 4.60·27-s − 1.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.08·3-s + 0.5·4-s + 1.98·5-s + 0.770·6-s − 0.448·7-s − 0.353·8-s + 0.186·9-s − 1.40·10-s − 1.13·11-s − 0.544·12-s + 0.0364·13-s + 0.316·14-s − 2.16·15-s + 0.250·16-s + 1.18·17-s − 0.131·18-s + 1.00·19-s + 0.993·20-s + 0.488·21-s + 0.805·22-s + 0.188·23-s + 0.385·24-s + 2.94·25-s − 0.0257·26-s + 0.886·27-s − 0.224·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214961396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214961396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 1.88T + 3T^{2} \) |
| 5 | \( 1 - 4.44T + 5T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 - 0.131T + 13T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 - 0.903T + 23T^{2} \) |
| 29 | \( 1 + 6.51T + 29T^{2} \) |
| 37 | \( 1 - 8.11T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 + 7.13T + 43T^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 + 4.02T + 71T^{2} \) |
| 73 | \( 1 - 1.74T + 73T^{2} \) |
| 79 | \( 1 + 0.522T + 79T^{2} \) |
| 83 | \( 1 + 9.56T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
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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
−8.136545188220533324400760591482, −7.17119447032242742357687407659, −6.60934233762557918595980432797, −5.79106261036448864160979770633, −5.47646724214581791124505901266, −5.04377384675993053992423676230, −3.30419653851187011615676110959, −2.58834419491428949285272196250, −1.63839313273736646481247887727, −0.69675869221756755621916584002,
0.69675869221756755621916584002, 1.63839313273736646481247887727, 2.58834419491428949285272196250, 3.30419653851187011615676110959, 5.04377384675993053992423676230, 5.47646724214581791124505901266, 5.79106261036448864160979770633, 6.60934233762557918595980432797, 7.17119447032242742357687407659, 8.136545188220533324400760591482
