| L(s) = 1 | + 1.20·2-s − 0.150·3-s − 0.537·4-s − 3.32·5-s − 0.181·6-s − 3.06·8-s − 2.97·9-s − 4.02·10-s + 2.93·11-s + 0.0807·12-s − 1.58·13-s + 0.500·15-s − 2.63·16-s − 1.28·17-s − 3.60·18-s − 2.64·19-s + 1.78·20-s + 3.55·22-s + 7.50·23-s + 0.461·24-s + 6.07·25-s − 1.91·26-s + 0.899·27-s − 0.257·29-s + 0.605·30-s + 7.18·31-s + 2.94·32-s + ⋯ |
| L(s) = 1 | + 0.855·2-s − 0.0868·3-s − 0.268·4-s − 1.48·5-s − 0.0742·6-s − 1.08·8-s − 0.992·9-s − 1.27·10-s + 0.886·11-s + 0.0233·12-s − 0.439·13-s + 0.129·15-s − 0.659·16-s − 0.311·17-s − 0.848·18-s − 0.605·19-s + 0.399·20-s + 0.758·22-s + 1.56·23-s + 0.0942·24-s + 1.21·25-s − 0.375·26-s + 0.173·27-s − 0.0478·29-s + 0.110·30-s + 1.29·31-s + 0.520·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174027358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174027358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 3 | \( 1 + 0.150T + 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 + 0.257T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 - 9.78T + 59T^{2} \) |
| 61 | \( 1 + 4.88T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 + 9.76T + 71T^{2} \) |
| 73 | \( 1 - 4.95T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 7.63T + 89T^{2} \) |
| 97 | \( 1 - 3.42T + 97T^{2} \) |
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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
−8.780466416527298233152754404591, −8.602621906766477084059123417630, −7.54878704097124901789051626417, −6.66892490180910410976157048054, −5.89998985046388087553574838085, −4.79681376925541073822605426807, −4.35981019749083271793097618162, −3.44834488370575579912638021674, −2.74573667381795119864368558509, −0.62066947478863401049904423825,
0.62066947478863401049904423825, 2.74573667381795119864368558509, 3.44834488370575579912638021674, 4.35981019749083271793097618162, 4.79681376925541073822605426807, 5.89998985046388087553574838085, 6.66892490180910410976157048054, 7.54878704097124901789051626417, 8.602621906766477084059123417630, 8.780466416527298233152754404591
