| L(s) = 1 | + 4.60·2-s + 13.1·4-s + 7·7-s + 23.8·8-s + 52.9·11-s + 19.6·13-s + 32.2·14-s + 4.19·16-s − 61.5·17-s + 27.0·19-s + 243.·22-s − 19.2·23-s + 90.2·26-s + 92.2·28-s + 167.·29-s + 225.·31-s − 171.·32-s − 283.·34-s + 311.·37-s + 124.·38-s − 12.8·41-s − 114.·43-s + 697.·44-s − 88.4·46-s + 207.·47-s + 49·49-s + 258.·52-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.64·4-s + 0.377·7-s + 1.05·8-s + 1.45·11-s + 0.418·13-s + 0.614·14-s + 0.0655·16-s − 0.877·17-s + 0.327·19-s + 2.36·22-s − 0.174·23-s + 0.680·26-s + 0.622·28-s + 1.07·29-s + 1.30·31-s − 0.945·32-s − 1.42·34-s + 1.38·37-s + 0.532·38-s − 0.0487·41-s − 0.405·43-s + 2.39·44-s − 0.283·46-s + 0.645·47-s + 0.142·49-s + 0.688·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.046461268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.046461268\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.60T + 8T^{2} \) |
| 11 | \( 1 - 52.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 311.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 12.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 114.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 207.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 227.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 605.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 315.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 720.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 56.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 692.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 661.T + 9.12e5T^{2} \) |
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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
−8.999793589605456452800821061686, −8.208718184942912046615153386147, −6.99518441399130246247161077767, −6.43426336740754356236685980547, −5.76550558178173522107636173244, −4.62179352516964987083629579962, −4.23862940236413818815328166314, −3.26899224861962938088793056437, −2.26969771098484712920137327649, −1.07461484927796161767912244916,
1.07461484927796161767912244916, 2.26969771098484712920137327649, 3.26899224861962938088793056437, 4.23862940236413818815328166314, 4.62179352516964987083629579962, 5.76550558178173522107636173244, 6.43426336740754356236685980547, 6.99518441399130246247161077767, 8.208718184942912046615153386147, 8.999793589605456452800821061686
