| L(s) = 1 | − 2.08·2-s + 2.34·4-s − 0.992·7-s − 0.726·8-s − 2·11-s + 3.37·13-s + 2.06·14-s − 3.18·16-s + 2.89·17-s − 2.58·19-s + 4.17·22-s − 4.54·23-s − 7.03·26-s − 2.33·28-s + 5.38·29-s + 0.136·31-s + 8.08·32-s − 6.03·34-s + 2.14·37-s + 5.38·38-s − 8.63·41-s − 4.64·43-s − 4.69·44-s + 9.48·46-s + 9.92·47-s − 6.01·49-s + 7.92·52-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.17·4-s − 0.375·7-s − 0.256·8-s − 0.603·11-s + 0.935·13-s + 0.553·14-s − 0.795·16-s + 0.702·17-s − 0.592·19-s + 0.889·22-s − 0.948·23-s − 1.37·26-s − 0.440·28-s + 0.999·29-s + 0.0245·31-s + 1.42·32-s − 1.03·34-s + 0.353·37-s + 0.874·38-s − 1.34·41-s − 0.708·43-s − 0.708·44-s + 1.39·46-s + 1.44·47-s − 0.859·49-s + 1.09·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 7 | \( 1 + 0.992T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 4.54T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 0.136T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 + 8.63T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 + 9.64T + 71T^{2} \) |
| 73 | \( 1 + 0.775T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−8.013349120409017845996347857977, −7.31822305758212588567333353962, −6.50988759556435075582067385179, −5.94142580597378125862878492063, −4.91485728550343781035436974867, −3.97193236324559571527278580574, −3.01569087737105795653655974068, −2.04855885656078612067051512535, −1.10552392187807172596628266991, 0,
1.10552392187807172596628266991, 2.04855885656078612067051512535, 3.01569087737105795653655974068, 3.97193236324559571527278580574, 4.91485728550343781035436974867, 5.94142580597378125862878492063, 6.50988759556435075582067385179, 7.31822305758212588567333353962, 8.013349120409017845996347857977
