| L(s) = 1 | − 1.52·2-s − 3-s + 0.313·4-s − 5-s + 1.52·6-s − 2.49·7-s + 2.56·8-s + 9-s + 1.52·10-s − 3.05·11-s − 0.313·12-s + 0.142·13-s + 3.79·14-s + 15-s − 4.52·16-s + 3.56·17-s − 1.52·18-s − 6.21·19-s − 0.313·20-s + 2.49·21-s + 4.65·22-s − 8.69·23-s − 2.56·24-s + 25-s − 0.216·26-s − 27-s − 0.781·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 0.577·3-s + 0.156·4-s − 0.447·5-s + 0.620·6-s − 0.943·7-s + 0.906·8-s + 0.333·9-s + 0.480·10-s − 0.922·11-s − 0.0904·12-s + 0.0394·13-s + 1.01·14-s + 0.258·15-s − 1.13·16-s + 0.863·17-s − 0.358·18-s − 1.42·19-s − 0.0700·20-s + 0.544·21-s + 0.991·22-s − 1.81·23-s − 0.523·24-s + 0.200·25-s − 0.0423·26-s − 0.192·27-s − 0.147·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09674522862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09674522862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 - 0.142T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 6.21T + 19T^{2} \) |
| 23 | \( 1 + 8.69T + 23T^{2} \) |
| 29 | \( 1 + 0.813T + 29T^{2} \) |
| 31 | \( 1 - 5.86T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 - 0.462T + 41T^{2} \) |
| 43 | \( 1 + 5.58T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 - 7.14T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 0.423T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.037822930536059748745409525888, −7.69214385135548779122895878038, −6.65096174985150120930958395047, −6.21460576239827991784367034850, −5.23788551686314132390661092002, −4.43780544413618396656336659599, −3.70574223834835165917414081741, −2.61710921703321333101715337094, −1.52219591779394760967927949433, −0.20142196609897231250121746107,
0.20142196609897231250121746107, 1.52219591779394760967927949433, 2.61710921703321333101715337094, 3.70574223834835165917414081741, 4.43780544413618396656336659599, 5.23788551686314132390661092002, 6.21460576239827991784367034850, 6.65096174985150120930958395047, 7.69214385135548779122895878038, 8.037822930536059748745409525888
