| L(s) = 1 | + 3·3-s − 10.5·5-s − 7·7-s + 9·9-s + 34.7·11-s + 37.2·13-s − 31.6·15-s − 10.5·17-s − 58.5·19-s − 21·21-s + 125.·23-s − 13.7·25-s + 27·27-s + 35.4·29-s − 291.·31-s + 104.·33-s + 73.8·35-s + 259.·37-s + 111.·39-s − 338.·41-s + 6.80·43-s − 94.9·45-s − 250.·47-s + 49·49-s − 31.6·51-s + 536.·53-s − 366.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.943·5-s − 0.377·7-s + 0.333·9-s + 0.952·11-s + 0.795·13-s − 0.544·15-s − 0.150·17-s − 0.707·19-s − 0.218·21-s + 1.13·23-s − 0.109·25-s + 0.192·27-s + 0.226·29-s − 1.69·31-s + 0.549·33-s + 0.356·35-s + 1.15·37-s + 0.459·39-s − 1.28·41-s + 0.0241·43-s − 0.314·45-s − 0.778·47-s + 0.142·49-s − 0.0868·51-s + 1.39·53-s − 0.898·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.131360086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131360086\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−9.006743623146583624947976169006, −8.585709901700530857523540634968, −7.64373570667051806031404691358, −6.88863218169885432462384766345, −6.11866534038416535760956524620, −4.80942102053493307172613486536, −3.81037685308484063842822231186, −3.39639940542760708770362563503, −1.99491353710157120667241029432, −0.71519488616983047452452648080,
0.71519488616983047452452648080, 1.99491353710157120667241029432, 3.39639940542760708770362563503, 3.81037685308484063842822231186, 4.80942102053493307172613486536, 6.11866534038416535760956524620, 6.88863218169885432462384766345, 7.64373570667051806031404691358, 8.585709901700530857523540634968, 9.006743623146583624947976169006
