| L(s) = 1 | + 3-s + 2.45·7-s + 9-s + 3.77·11-s + 13-s + 0.793·17-s + 3.15·19-s + 2.45·21-s + 1.20·23-s + 27-s + 5.04·29-s − 4.94·31-s + 3.77·33-s + 9.91·37-s + 39-s − 10.6·41-s − 6.58·43-s + 9.60·47-s − 0.951·49-s + 0.793·51-s − 0.168·53-s + 3.15·57-s + 12.0·59-s + 15.2·61-s + 2.45·63-s − 3.33·67-s + 1.20·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.929·7-s + 0.333·9-s + 1.13·11-s + 0.277·13-s + 0.192·17-s + 0.724·19-s + 0.536·21-s + 0.251·23-s + 0.192·27-s + 0.936·29-s − 0.888·31-s + 0.657·33-s + 1.62·37-s + 0.160·39-s − 1.65·41-s − 1.00·43-s + 1.40·47-s − 0.135·49-s + 0.111·51-s − 0.0231·53-s + 0.418·57-s + 1.56·59-s + 1.95·61-s + 0.309·63-s − 0.407·67-s + 0.145·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.574515045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.574515045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 17 | \( 1 - 0.793T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 4.94T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 - 9.60T + 47T^{2} \) |
| 53 | \( 1 + 0.168T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 6.46T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 - 1.89T + 89T^{2} \) |
| 97 | \( 1 - 9.62T + 97T^{2} \) |
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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
−7.960314465311518278924263928096, −7.18174479002653658560297868750, −6.64441265086728356039812671343, −5.71334347194642693277400159650, −4.99231266058187858668201491890, −4.20913331195260858157080142417, −3.58206925221124993070935835242, −2.69055034946888431935352712086, −1.68775744121820242039384129932, −1.01360122207787873070260412738,
1.01360122207787873070260412738, 1.68775744121820242039384129932, 2.69055034946888431935352712086, 3.58206925221124993070935835242, 4.20913331195260858157080142417, 4.99231266058187858668201491890, 5.71334347194642693277400159650, 6.64441265086728356039812671343, 7.18174479002653658560297868750, 7.960314465311518278924263928096
