| L(s) = 1 | + 5.38·2-s + 20.9·4-s − 7·7-s + 69.8·8-s + 15.8·11-s + 40.8·13-s − 37.6·14-s + 208.·16-s − 47.2·17-s + 141.·19-s + 85.5·22-s + 97.3·23-s + 219.·26-s − 146.·28-s − 236.·29-s + 95.7·31-s + 562.·32-s − 254.·34-s − 33.8·37-s + 759.·38-s − 197.·41-s − 437.·43-s + 333.·44-s + 523.·46-s + 256.·47-s + 49·49-s + 857.·52-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.62·4-s − 0.377·7-s + 3.08·8-s + 0.435·11-s + 0.871·13-s − 0.719·14-s + 3.25·16-s − 0.674·17-s + 1.70·19-s + 0.828·22-s + 0.882·23-s + 1.65·26-s − 0.991·28-s − 1.51·29-s + 0.555·31-s + 3.10·32-s − 1.28·34-s − 0.150·37-s + 3.24·38-s − 0.751·41-s − 1.55·43-s + 1.14·44-s + 1.67·46-s + 0.796·47-s + 0.142·49-s + 2.28·52-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
| \(L(2)\) |
\(\approx\) |
\(8.886723156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.886723156\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.38T + 8T^{2} \) |
| 11 | \( 1 - 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 97.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 95.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 33.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 256.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 64.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 809.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 318.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 726.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 68.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 680.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 317.T + 9.12e5T^{2} \) |
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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
−9.109507087618428990309508211530, −7.950148855167651331101153224910, −6.93672855782459283436468769434, −6.56282210444345660115698833718, −5.51599942164465942862723171174, −5.03774825413436448632876541476, −3.80876620511970067230560663887, −3.44560782723937798265043854504, −2.34005566229234304097251827457, −1.18148071168082199611342748806,
1.18148071168082199611342748806, 2.34005566229234304097251827457, 3.44560782723937798265043854504, 3.80876620511970067230560663887, 5.03774825413436448632876541476, 5.51599942164465942862723171174, 6.56282210444345660115698833718, 6.93672855782459283436468769434, 7.950148855167651331101153224910, 9.109507087618428990309508211530
