| L(s) = 1 | + 1.80·2-s − 0.445·3-s + 1.24·4-s + 0.356·5-s − 0.801·6-s − 4.04·7-s − 1.35·8-s − 2.80·9-s + 0.643·10-s − 2.91·11-s − 0.554·12-s + 5.18·13-s − 7.29·14-s − 0.158·15-s − 4.93·16-s − 1.10·17-s − 5.04·18-s − 2.04·19-s + 0.445·20-s + 1.80·21-s − 5.24·22-s − 4.13·23-s + 0.603·24-s − 4.87·25-s + 9.34·26-s + 2.58·27-s − 5.04·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 0.256·3-s + 0.623·4-s + 0.159·5-s − 0.327·6-s − 1.53·7-s − 0.479·8-s − 0.933·9-s + 0.203·10-s − 0.877·11-s − 0.160·12-s + 1.43·13-s − 1.94·14-s − 0.0410·15-s − 1.23·16-s − 0.269·17-s − 1.19·18-s − 0.470·19-s + 0.0995·20-s + 0.393·21-s − 1.11·22-s − 0.862·23-s + 0.123·24-s − 0.974·25-s + 1.83·26-s + 0.496·27-s − 0.954·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 3 | \( 1 + 0.445T + 3T^{2} \) |
| 5 | \( 1 - 0.356T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 + 2.04T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 - 2.91T + 37T^{2} \) |
| 41 | \( 1 + 0.396T + 41T^{2} \) |
| 43 | \( 1 + 5.74T + 43T^{2} \) |
| 47 | \( 1 - 7.80T + 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 - 0.374T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 - 0.594T + 79T^{2} \) |
| 83 | \( 1 + 9.43T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−9.888248606523496886580597135121, −8.950124711432689622074153127905, −8.107137431893207583710431436740, −6.57056055058688933030254316122, −6.09688110311587608278971942747, −5.52908969738945855010800836708, −4.24688492911943017040765548776, −3.36770041771616000057194031009, −2.56346244963340507972511034433, 0,
2.56346244963340507972511034433, 3.36770041771616000057194031009, 4.24688492911943017040765548776, 5.52908969738945855010800836708, 6.09688110311587608278971942747, 6.57056055058688933030254316122, 8.107137431893207583710431436740, 8.950124711432689622074153127905, 9.888248606523496886580597135121
