| L(s) = 1 | + 2.57·2-s + 0.279·3-s + 4.64·4-s + 1.29·5-s + 0.720·6-s + 7-s + 6.80·8-s − 2.92·9-s + 3.34·10-s − 11-s + 1.29·12-s − 13-s + 2.57·14-s + 0.362·15-s + 8.26·16-s + 5.85·17-s − 7.53·18-s − 1.34·19-s + 6.02·20-s + 0.279·21-s − 2.57·22-s + 3.21·23-s + 1.90·24-s − 3.31·25-s − 2.57·26-s − 1.65·27-s + 4.64·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.161·3-s + 2.32·4-s + 0.580·5-s + 0.294·6-s + 0.377·7-s + 2.40·8-s − 0.973·9-s + 1.05·10-s − 0.301·11-s + 0.374·12-s − 0.277·13-s + 0.688·14-s + 0.0936·15-s + 2.06·16-s + 1.42·17-s − 1.77·18-s − 0.308·19-s + 1.34·20-s + 0.0610·21-s − 0.549·22-s + 0.670·23-s + 0.388·24-s − 0.663·25-s − 0.505·26-s − 0.318·27-s + 0.877·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.254906749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.254906749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 - 0.279T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 - 3.21T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 + 0.487T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 + 9.51T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 + 1.50T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 6.95T + 61T^{2} \) |
| 67 | \( 1 + 6.84T + 67T^{2} \) |
| 71 | \( 1 + 3.11T + 71T^{2} \) |
| 73 | \( 1 - 2.04T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 0.0832T + 83T^{2} \) |
| 89 | \( 1 + 5.48T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−10.27292543713285894554250595913, −9.202334462812602146222166703594, −7.999226497968244274200313029165, −7.24298955476015404178760561284, −6.07808905235355050281259159183, −5.56600034044730814653062421551, −4.87848805028244659024304438165, −3.67067626144314623551527579093, −2.85913010039949438828096232929, −1.85384398790485958740322145785,
1.85384398790485958740322145785, 2.85913010039949438828096232929, 3.67067626144314623551527579093, 4.87848805028244659024304438165, 5.56600034044730814653062421551, 6.07808905235355050281259159183, 7.24298955476015404178760561284, 7.999226497968244274200313029165, 9.202334462812602146222166703594, 10.27292543713285894554250595913
