| L(s) = 1 | + 2.51·2-s − 3.29·3-s + 4.34·4-s + 5-s − 8.28·6-s + 5.90·8-s + 7.83·9-s + 2.51·10-s − 11-s − 14.2·12-s + 5.30·13-s − 3.29·15-s + 6.18·16-s + 1.79·17-s + 19.7·18-s − 5.49·19-s + 4.34·20-s − 2.51·22-s − 0.509·23-s − 19.4·24-s + 25-s + 13.3·26-s − 15.8·27-s + 4.77·29-s − 8.28·30-s + 5.61·31-s + 3.76·32-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 1.90·3-s + 2.17·4-s + 0.447·5-s − 3.38·6-s + 2.08·8-s + 2.61·9-s + 0.796·10-s − 0.301·11-s − 4.12·12-s + 1.47·13-s − 0.849·15-s + 1.54·16-s + 0.434·17-s + 4.64·18-s − 1.25·19-s + 0.971·20-s − 0.536·22-s − 0.106·23-s − 3.96·24-s + 0.200·25-s + 2.61·26-s − 3.05·27-s + 0.886·29-s − 1.51·30-s + 1.00·31-s + 0.665·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.520467304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.520467304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 + 3.29T + 3T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 23 | \( 1 + 0.509T + 23T^{2} \) |
| 29 | \( 1 - 4.77T + 29T^{2} \) |
| 31 | \( 1 - 5.61T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 - 6.22T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 + 5.80T + 59T^{2} \) |
| 61 | \( 1 - 1.94T + 61T^{2} \) |
| 67 | \( 1 - 1.77T + 67T^{2} \) |
| 71 | \( 1 + 3.97T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 - 4.07T + 83T^{2} \) |
| 89 | \( 1 - 3.11T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.825047871837085320428717042726, −7.56476257485122508707718033379, −6.66111862670060348209875354833, −6.15890796300928679373730141849, −5.81486209707375404588705991203, −4.98213417687674174875513796580, −4.37911205411294505252583368325, −3.59597028535846707553561585208, −2.21668960235290726405892461421, −1.04610837227610474902528868308,
1.04610837227610474902528868308, 2.21668960235290726405892461421, 3.59597028535846707553561585208, 4.37911205411294505252583368325, 4.98213417687674174875513796580, 5.81486209707375404588705991203, 6.15890796300928679373730141849, 6.66111862670060348209875354833, 7.56476257485122508707718033379, 8.825047871837085320428717042726
