| L(s) = 1 | + 7·3-s + 7·5-s − 13·7-s + 22·9-s + 26·11-s − 13·13-s + 49·15-s + 77·17-s + 126·19-s − 91·21-s − 96·23-s − 76·25-s − 35·27-s + 82·29-s + 196·31-s + 182·33-s − 91·35-s + 131·37-s − 91·39-s + 336·41-s + 201·43-s + 154·45-s − 105·47-s − 174·49-s + 539·51-s + 432·53-s + 182·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 0.626·5-s − 0.701·7-s + 0.814·9-s + 0.712·11-s − 0.277·13-s + 0.843·15-s + 1.09·17-s + 1.52·19-s − 0.945·21-s − 0.870·23-s − 0.607·25-s − 0.249·27-s + 0.525·29-s + 1.13·31-s + 0.960·33-s − 0.439·35-s + 0.582·37-s − 0.373·39-s + 1.27·41-s + 0.712·43-s + 0.510·45-s − 0.325·47-s − 0.507·49-s + 1.47·51-s + 1.11·53-s + 0.446·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.875475822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.875475822\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 26 T + p^{3} T^{2} \) |
| 17 | \( 1 - 77 T + p^{3} T^{2} \) |
| 19 | \( 1 - 126 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 - 131 T + p^{3} T^{2} \) |
| 41 | \( 1 - 336 T + p^{3} T^{2} \) |
| 43 | \( 1 - 201 T + p^{3} T^{2} \) |
| 47 | \( 1 + 105 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 294 T + p^{3} T^{2} \) |
| 61 | \( 1 - 56 T + p^{3} T^{2} \) |
| 67 | \( 1 + 478 T + p^{3} T^{2} \) |
| 71 | \( 1 - 9 T + p^{3} T^{2} \) |
| 73 | \( 1 - 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1304 T + p^{3} T^{2} \) |
| 83 | \( 1 - 308 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−9.659427596425168850646811232214, −9.204884160311161631402700884428, −8.103408709772233408857181326977, −7.49779765513871737451526477627, −6.36674168571903000928125249596, −5.52371354694228950242423558095, −4.07775398956517935348391579653, −3.20416372343305085432037383032, −2.39050417149120808523245837153, −1.09307455589565350870158210031,
1.09307455589565350870158210031, 2.39050417149120808523245837153, 3.20416372343305085432037383032, 4.07775398956517935348391579653, 5.52371354694228950242423558095, 6.36674168571903000928125249596, 7.49779765513871737451526477627, 8.103408709772233408857181326977, 9.204884160311161631402700884428, 9.659427596425168850646811232214
