L(s) = 1 | + 2.05·2-s − 2.28·3-s + 2.21·4-s + 0.331·5-s − 4.68·6-s + 3.06·7-s + 0.444·8-s + 2.20·9-s + 0.679·10-s + 1.18·11-s − 5.05·12-s + 13-s + 6.30·14-s − 0.755·15-s − 3.52·16-s + 3.46·17-s + 4.52·18-s − 1.47·19-s + 0.733·20-s − 6.99·21-s + 2.43·22-s + 4.02·23-s − 1.01·24-s − 4.89·25-s + 2.05·26-s + 1.81·27-s + 6.80·28-s + ⋯ |
L(s) = 1 | + 1.45·2-s − 1.31·3-s + 1.10·4-s + 0.148·5-s − 1.91·6-s + 1.15·7-s + 0.157·8-s + 0.734·9-s + 0.214·10-s + 0.357·11-s − 1.45·12-s + 0.277·13-s + 1.68·14-s − 0.194·15-s − 0.880·16-s + 0.840·17-s + 1.06·18-s − 0.338·19-s + 0.164·20-s − 1.52·21-s + 0.519·22-s + 0.838·23-s − 0.206·24-s − 0.978·25-s + 0.402·26-s + 0.349·27-s + 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.735290382\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.735290382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 3 | \( 1 + 2.28T + 3T^{2} \) |
| 5 | \( 1 - 0.331T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 - 4.02T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 - 5.32T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 + 0.520T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 - 2.51T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 + 2.03T + 59T^{2} \) |
| 61 | \( 1 - 3.54T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 0.954T + 79T^{2} \) |
| 83 | \( 1 - 3.48T + 83T^{2} \) |
| 89 | \( 1 + 1.29T + 89T^{2} \) |
| 97 | \( 1 + 0.434T + 97T^{2} \) |
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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
−9.919975523821179763680256535743, −8.729210803776518710473146039472, −7.75037499042932397119920596645, −6.64117431994894675418180621599, −6.03809323080150615562899606169, −5.31281527113416974298945318094, −4.72403836609189739959587329264, −3.95308405699535851572683503472, −2.61578578013696519768690682648, −1.12495505966587508110403174908,
1.12495505966587508110403174908, 2.61578578013696519768690682648, 3.95308405699535851572683503472, 4.72403836609189739959587329264, 5.31281527113416974298945318094, 6.03809323080150615562899606169, 6.64117431994894675418180621599, 7.75037499042932397119920596645, 8.729210803776518710473146039472, 9.919975523821179763680256535743