| L(s) = 1 | − 0.414·2-s − 2.61·3-s − 1.82·4-s − 1.84·5-s + 1.08·6-s − 1.08·7-s + 1.58·8-s + 3.82·9-s + 0.765·10-s + 1.08·11-s + 4.77·12-s − 1.41·13-s + 0.448·14-s + 4.82·15-s + 3·16-s − 1.58·18-s + 4.82·19-s + 3.37·20-s + 2.82·21-s − 0.448·22-s + 4.14·23-s − 4.14·24-s − 1.58·25-s + 0.585·26-s − 2.16·27-s + 1.97·28-s − 4.46·29-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 1.50·3-s − 0.914·4-s − 0.826·5-s + 0.441·6-s − 0.409·7-s + 0.560·8-s + 1.27·9-s + 0.242·10-s + 0.326·11-s + 1.37·12-s − 0.392·13-s + 0.119·14-s + 1.24·15-s + 0.750·16-s − 0.373·18-s + 1.10·19-s + 0.755·20-s + 0.617·21-s − 0.0955·22-s + 0.864·23-s − 0.845·24-s − 0.317·25-s + 0.114·26-s − 0.416·27-s + 0.374·28-s − 0.828·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3709860928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3709860928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 - 1.08T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 - 8.15T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 9.23T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 - 0.343T + 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 - 6.43T + 97T^{2} \) |
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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
−11.77428633826789162191233649121, −10.97301281474209944627203648970, −9.980925988435145467801198055134, −9.145423464061301604800758181075, −7.84082660741091171755069412350, −6.92119285087509769653199325927, −5.64341057722199226706860081028, −4.79888709504384833104850093883, −3.68342681456816205940596894245, −0.71053221650369498753818566640,
0.71053221650369498753818566640, 3.68342681456816205940596894245, 4.79888709504384833104850093883, 5.64341057722199226706860081028, 6.92119285087509769653199325927, 7.84082660741091171755069412350, 9.145423464061301604800758181075, 9.980925988435145467801198055134, 10.97301281474209944627203648970, 11.77428633826789162191233649121
