| L(s) = 1 | − 0.739·2-s − 0.0660·3-s − 1.45·4-s + 3.87·5-s + 0.0488·6-s + 2.55·8-s − 2.99·9-s − 2.86·10-s + 2.16·11-s + 0.0959·12-s − 0.256·15-s + 1.01·16-s − 2.63·17-s + 2.21·18-s − 6.29·19-s − 5.63·20-s − 1.59·22-s + 2.42·23-s − 0.168·24-s + 10.0·25-s + 0.395·27-s − 3.40·29-s + 0.189·30-s − 2.27·31-s − 5.85·32-s − 0.142·33-s + 1.94·34-s + ⋯ |
| L(s) = 1 | − 0.522·2-s − 0.0381·3-s − 0.726·4-s + 1.73·5-s + 0.0199·6-s + 0.902·8-s − 0.998·9-s − 0.907·10-s + 0.652·11-s + 0.0277·12-s − 0.0661·15-s + 0.254·16-s − 0.638·17-s + 0.522·18-s − 1.44·19-s − 1.26·20-s − 0.340·22-s + 0.506·23-s − 0.0344·24-s + 2.00·25-s + 0.0762·27-s − 0.632·29-s + 0.0345·30-s − 0.407·31-s − 1.03·32-s − 0.0248·33-s + 0.333·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.443171612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443171612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.739T + 2T^{2} \) |
| 3 | \( 1 + 0.0660T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 + 2.27T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 0.822T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 2.21T + 89T^{2} \) |
| 97 | \( 1 - 6.04T + 97T^{2} \) |
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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
−8.132207379068423703205553817921, −6.85048301697788630290622508865, −6.57410775328093380346989118925, −5.50675587428343287905922167325, −5.35789869239066079602237896239, −4.33885414571121666944271367500, −3.49042507768372288940743195150, −2.30117798016938183613878610609, −1.82230673226260972103961114314, −0.64158133609814132272947527706,
0.64158133609814132272947527706, 1.82230673226260972103961114314, 2.30117798016938183613878610609, 3.49042507768372288940743195150, 4.33885414571121666944271367500, 5.35789869239066079602237896239, 5.50675587428343287905922167325, 6.57410775328093380346989118925, 6.85048301697788630290622508865, 8.132207379068423703205553817921
