| L(s) = 1 | + 2.36·2-s + 2.46·3-s + 3.60·4-s − 3.97·5-s + 5.83·6-s − 0.131·7-s + 3.79·8-s + 3.08·9-s − 9.40·10-s + 2.70·11-s + 8.88·12-s − 3.80·13-s − 0.310·14-s − 9.80·15-s + 1.77·16-s + 0.111·17-s + 7.30·18-s − 1.29·19-s − 14.3·20-s − 0.323·21-s + 6.40·22-s + 6.25·23-s + 9.36·24-s + 10.7·25-s − 9.00·26-s + 0.212·27-s − 0.472·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.42·3-s + 1.80·4-s − 1.77·5-s + 2.38·6-s − 0.0495·7-s + 1.34·8-s + 1.02·9-s − 2.97·10-s + 0.815·11-s + 2.56·12-s − 1.05·13-s − 0.0829·14-s − 2.53·15-s + 0.443·16-s + 0.0269·17-s + 1.72·18-s − 0.296·19-s − 3.20·20-s − 0.0705·21-s + 1.36·22-s + 1.30·23-s + 1.91·24-s + 2.15·25-s − 1.76·26-s + 0.0408·27-s − 0.0892·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.721469963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.721469963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 - T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 3 | \( 1 - 2.46T + 3T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 + 0.131T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 - 0.111T + 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 - 6.25T + 23T^{2} \) |
| 29 | \( 1 + 7.45T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 8.75T + 37T^{2} \) |
| 41 | \( 1 - 1.70T + 41T^{2} \) |
| 43 | \( 1 + 5.20T + 43T^{2} \) |
| 47 | \( 1 + 0.408T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 4.28T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 4.35T + 97T^{2} \) |
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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
−11.65129461305059935391034473176, −11.19447354487721694072128520071, −9.466821424736877686468874712420, −8.489303452897870940009841857085, −7.44911908711757695911530131584, −6.92043405397958436319411973388, −5.14251319797084118865000546198, −4.00312325504471041821130605154, −3.59502851724776522399679665107, −2.48782597627328057677888504330,
2.48782597627328057677888504330, 3.59502851724776522399679665107, 4.00312325504471041821130605154, 5.14251319797084118865000546198, 6.92043405397958436319411973388, 7.44911908711757695911530131584, 8.489303452897870940009841857085, 9.466821424736877686468874712420, 11.19447354487721694072128520071, 11.65129461305059935391034473176
