L(s) = 1 | + 5·2-s − 7·3-s + 17·4-s + 7·5-s − 35·6-s + 13·7-s + 45·8-s + 22·9-s + 35·10-s + 26·11-s − 119·12-s + 65·14-s − 49·15-s + 89·16-s + 77·17-s + 110·18-s + 126·19-s + 119·20-s − 91·21-s + 130·22-s − 96·23-s − 315·24-s − 76·25-s + 35·27-s + 221·28-s − 82·29-s − 245·30-s + ⋯ |
L(s) = 1 | + 1.76·2-s − 1.34·3-s + 17/8·4-s + 0.626·5-s − 2.38·6-s + 0.701·7-s + 1.98·8-s + 0.814·9-s + 1.10·10-s + 0.712·11-s − 2.86·12-s + 1.24·14-s − 0.843·15-s + 1.39·16-s + 1.09·17-s + 1.44·18-s + 1.52·19-s + 1.33·20-s − 0.945·21-s + 1.25·22-s − 0.870·23-s − 2.67·24-s − 0.607·25-s + 0.249·27-s + 1.49·28-s − 0.525·29-s − 1.49·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.741751828\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.741751828\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 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
−12.06982900588716248438967969865, −11.80434401370175741148290795958, −10.90742058065693076721969271373, −9.716506956626598909344307784643, −7.58851824789922030946101308103, −6.37084450086936827568846446539, −5.58744463371738230345339955750, −4.97680026459199100254139694471, −3.57335445455789660377818728774, −1.61421828074687945911669584549,
1.61421828074687945911669584549, 3.57335445455789660377818728774, 4.97680026459199100254139694471, 5.58744463371738230345339955750, 6.37084450086936827568846446539, 7.58851824789922030946101308103, 9.716506956626598909344307784643, 10.90742058065693076721969271373, 11.80434401370175741148290795958, 12.06982900588716248438967969865