| L(s) = 1 | + 1.26·2-s − 3·3-s − 6.40·4-s − 3.78·6-s + 13.4·7-s − 18.1·8-s + 9·9-s − 1.30·11-s + 19.2·12-s − 85.6·13-s + 16.9·14-s + 28.3·16-s − 67.9·17-s + 11.3·18-s − 63.3·19-s − 40.2·21-s − 1.65·22-s + 139.·23-s + 54.5·24-s − 108.·26-s − 27·27-s − 86.0·28-s − 29·29-s − 170.·31-s + 181.·32-s + 3.92·33-s − 85.7·34-s + ⋯ |
| L(s) = 1 | + 0.446·2-s − 0.577·3-s − 0.800·4-s − 0.257·6-s + 0.725·7-s − 0.803·8-s + 0.333·9-s − 0.0358·11-s + 0.462·12-s − 1.82·13-s + 0.323·14-s + 0.442·16-s − 0.969·17-s + 0.148·18-s − 0.764·19-s − 0.418·21-s − 0.0160·22-s + 1.26·23-s + 0.463·24-s − 0.815·26-s − 0.192·27-s − 0.580·28-s − 0.185·29-s − 0.987·31-s + 1.00·32-s + 0.0207·33-s − 0.432·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8086071464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8086071464\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 1.26T + 8T^{2} \) |
| 7 | \( 1 - 13.4T + 343T^{2} \) |
| 11 | \( 1 + 1.30T + 1.33e3T^{2} \) |
| 13 | \( 1 + 85.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 139.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 405.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 178.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 93.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 279.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 793.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 460.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 803.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 150.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 313.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 250.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 97.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−8.786568850833090631822600559298, −7.913850235727991062890264366939, −7.08778118511899735922486933055, −6.26916578129936983006817897178, −5.12874031946229942397197209275, −4.90034503301860996241850162277, −4.15358857463406538588547531104, −2.92353665686188288148102104664, −1.82030909306106183479348907670, −0.38399695828922064251396769239,
0.38399695828922064251396769239, 1.82030909306106183479348907670, 2.92353665686188288148102104664, 4.15358857463406538588547531104, 4.90034503301860996241850162277, 5.12874031946229942397197209275, 6.26916578129936983006817897178, 7.08778118511899735922486933055, 7.913850235727991062890264366939, 8.786568850833090631822600559298
