| L(s) = 1 | − 3-s + 5-s − 4.35·7-s + 9-s + 5.13·11-s − 5.81·13-s − 15-s − 0.0511·17-s − 4.35·19-s + 4.35·21-s − 5.31·23-s + 25-s − 27-s − 7.06·29-s + 0.667·31-s − 5.13·33-s − 4.35·35-s + 0.126·37-s + 5.81·39-s + 1.87·41-s + 3.32·43-s + 45-s − 8.98·47-s + 11.9·49-s + 0.0511·51-s + 9.65·53-s + 5.13·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.64·7-s + 0.333·9-s + 1.54·11-s − 1.61·13-s − 0.258·15-s − 0.0124·17-s − 0.997·19-s + 0.951·21-s − 1.10·23-s + 0.200·25-s − 0.192·27-s − 1.31·29-s + 0.119·31-s − 0.894·33-s − 0.736·35-s + 0.0208·37-s + 0.931·39-s + 0.292·41-s + 0.506·43-s + 0.149·45-s − 1.31·47-s + 1.71·49-s + 0.00716·51-s + 1.32·53-s + 0.692·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8090634527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8090634527\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 + 5.81T + 13T^{2} \) |
| 17 | \( 1 + 0.0511T + 17T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 + 5.31T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 - 0.667T + 31T^{2} \) |
| 37 | \( 1 - 0.126T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 - 3.32T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 + 0.775T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 71 | \( 1 + 7.85T + 71T^{2} \) |
| 73 | \( 1 - 8.18T + 73T^{2} \) |
| 79 | \( 1 - 1.89T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 + 4.35T + 97T^{2} \) |
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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
−7.57167126504814417044524193301, −6.94708949573752701344368124002, −6.37501699764258154573765684208, −5.99937366137867898680494472159, −5.13043597261969280838742541389, −4.16698877762956856621255044438, −3.67581096607998762721582188630, −2.59674609909080939603289186691, −1.81466316889829589176589664966, −0.43939665758109311471516992525,
0.43939665758109311471516992525, 1.81466316889829589176589664966, 2.59674609909080939603289186691, 3.67581096607998762721582188630, 4.16698877762956856621255044438, 5.13043597261969280838742541389, 5.99937366137867898680494472159, 6.37501699764258154573765684208, 6.94708949573752701344368124002, 7.57167126504814417044524193301
