| L(s) = 1 | − 0.193·2-s − 1.96·4-s − 3.35·7-s + 0.768·8-s − 11-s − 2.96·13-s + 0.649·14-s + 3.77·16-s − 4.57·17-s − 4.31·19-s + 0.193·22-s − 6.70·23-s + 0.574·26-s + 6.57·28-s + 3.61·29-s + 9.92·31-s − 2.26·32-s + 0.887·34-s + 2·37-s + 0.836·38-s + 4.38·41-s + 9.27·43-s + 1.96·44-s + 1.29·46-s − 9.92·47-s + 4.22·49-s + 5.81·52-s + ⋯ |
| L(s) = 1 | − 0.137·2-s − 0.981·4-s − 1.26·7-s + 0.271·8-s − 0.301·11-s − 0.821·13-s + 0.173·14-s + 0.943·16-s − 1.10·17-s − 0.989·19-s + 0.0413·22-s − 1.39·23-s + 0.112·26-s + 1.24·28-s + 0.670·29-s + 1.78·31-s − 0.401·32-s + 0.152·34-s + 0.328·37-s + 0.135·38-s + 0.685·41-s + 1.41·43-s + 0.295·44-s + 0.191·46-s − 1.44·47-s + 0.603·49-s + 0.806·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5414767902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5414767902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.38T + 41T^{2} \) |
| 43 | \( 1 - 9.27T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 + 5.92T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
| 97 | \( 1 + 0.0752T + 97T^{2} \) |
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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
−9.025897326995239827070345955448, −8.249279634338735514916731155355, −7.54487065004958476052937792256, −6.40665978546195629469866990454, −6.04825957625956048759198549546, −4.69813017101905642134716650735, −4.32036737446344089561028567884, −3.19152721554331398219805452977, −2.24929861665857050811999273130, −0.45525087172520054253600267706,
0.45525087172520054253600267706, 2.24929861665857050811999273130, 3.19152721554331398219805452977, 4.32036737446344089561028567884, 4.69813017101905642134716650735, 6.04825957625956048759198549546, 6.40665978546195629469866990454, 7.54487065004958476052937792256, 8.249279634338735514916731155355, 9.025897326995239827070345955448
