| L(s) = 1 | + 2-s + 3-s + 4-s + 0.268·5-s + 6-s + 3.42·7-s + 8-s + 9-s + 0.268·10-s + 11-s + 12-s + 1.26·13-s + 3.42·14-s + 0.268·15-s + 16-s + 0.867·17-s + 18-s + 6.18·19-s + 0.268·20-s + 3.42·21-s + 22-s − 8.86·23-s + 24-s − 4.92·25-s + 1.26·26-s + 27-s + 3.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.120·5-s + 0.408·6-s + 1.29·7-s + 0.353·8-s + 0.333·9-s + 0.0850·10-s + 0.301·11-s + 0.288·12-s + 0.351·13-s + 0.915·14-s + 0.0694·15-s + 0.250·16-s + 0.210·17-s + 0.235·18-s + 1.41·19-s + 0.0601·20-s + 0.747·21-s + 0.213·22-s − 1.84·23-s + 0.204·24-s − 0.985·25-s + 0.248·26-s + 0.192·27-s + 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.896520278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.896520278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 - 0.268T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 - 0.867T + 17T^{2} \) |
| 19 | \( 1 - 6.18T + 19T^{2} \) |
| 23 | \( 1 + 8.86T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 - 8.55T + 37T^{2} \) |
| 41 | \( 1 - 3.44T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 5.99T + 47T^{2} \) |
| 53 | \( 1 - 4.13T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 67 | \( 1 + 0.380T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 2.36T + 73T^{2} \) |
| 79 | \( 1 + 8.37T + 79T^{2} \) |
| 83 | \( 1 - 2.32T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.158871890567133197650360066815, −7.75120761061341265800574477428, −7.16719083416261549912934665877, −5.77677909022724063222602645491, −5.72262413435157540712069016117, −4.36406794725729034724955942154, −4.09564601022005236552420561109, −2.98942932835518122744083189083, −2.04687185457531416279492577786, −1.26139052765123108472750141313,
1.26139052765123108472750141313, 2.04687185457531416279492577786, 2.98942932835518122744083189083, 4.09564601022005236552420561109, 4.36406794725729034724955942154, 5.72262413435157540712069016117, 5.77677909022724063222602645491, 7.16719083416261549912934665877, 7.75120761061341265800574477428, 8.158871890567133197650360066815
