| L(s) = 1 | + 2.60·2-s + 3-s + 4.80·4-s − 1.50·5-s + 2.60·6-s + 7-s + 7.30·8-s + 9-s − 3.92·10-s + 0.0869·11-s + 4.80·12-s + 2.60·14-s − 1.50·15-s + 9.45·16-s + 6.49·17-s + 2.60·18-s + 5.29·19-s − 7.22·20-s + 21-s + 0.226·22-s − 5.20·23-s + 7.30·24-s − 2.73·25-s + 27-s + 4.80·28-s − 5.97·29-s − 3.92·30-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 0.577·3-s + 2.40·4-s − 0.673·5-s + 1.06·6-s + 0.377·7-s + 2.58·8-s + 0.333·9-s − 1.24·10-s + 0.0262·11-s + 1.38·12-s + 0.697·14-s − 0.388·15-s + 2.36·16-s + 1.57·17-s + 0.614·18-s + 1.21·19-s − 1.61·20-s + 0.218·21-s + 0.0483·22-s − 1.08·23-s + 1.49·24-s − 0.546·25-s + 0.192·27-s + 0.907·28-s − 1.10·29-s − 0.716·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.533423699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.533423699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 11 | \( 1 - 0.0869T + 11T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 - 2.00T + 31T^{2} \) |
| 37 | \( 1 - 9.52T + 37T^{2} \) |
| 41 | \( 1 + 6.43T + 41T^{2} \) |
| 43 | \( 1 - 6.81T + 43T^{2} \) |
| 47 | \( 1 + 1.83T + 47T^{2} \) |
| 53 | \( 1 + 3.28T + 53T^{2} \) |
| 59 | \( 1 - 2.67T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 1.29T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 5.86T + 97T^{2} \) |
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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
−7.946161842074656137747355962022, −7.82907594243180296439473009329, −7.06396354781327397311320239794, −5.99510843365833574517278382189, −5.47569125749442113574654707931, −4.60562955779184332531744456335, −3.83203872120800922779346171910, −3.36997078514080371020244312358, −2.46393649948588858298311997294, −1.37867504615749066915191165232,
1.37867504615749066915191165232, 2.46393649948588858298311997294, 3.36997078514080371020244312358, 3.83203872120800922779346171910, 4.60562955779184332531744456335, 5.47569125749442113574654707931, 5.99510843365833574517278382189, 7.06396354781327397311320239794, 7.82907594243180296439473009329, 7.946161842074656137747355962022
