L(s) = 1 | + 1.30·2-s − 3-s − 0.302·4-s − 1.30·6-s − 7-s − 3·8-s + 9-s − 11-s + 0.302·12-s + 2.30·13-s − 1.30·14-s − 3.30·16-s + 2.60·17-s + 1.30·18-s − 2.69·19-s + 21-s − 1.30·22-s + 5.60·23-s + 3·24-s + 3·26-s − 27-s + 0.302·28-s − 5.21·29-s + 2.39·31-s + 1.69·32-s + 33-s + 3.39·34-s + ⋯ |
L(s) = 1 | + 0.921·2-s − 0.577·3-s − 0.151·4-s − 0.531·6-s − 0.377·7-s − 1.06·8-s + 0.333·9-s − 0.301·11-s + 0.0874·12-s + 0.638·13-s − 0.348·14-s − 0.825·16-s + 0.631·17-s + 0.307·18-s − 0.618·19-s + 0.218·21-s − 0.277·22-s + 1.16·23-s + 0.612·24-s + 0.588·26-s − 0.192·27-s + 0.0572·28-s − 0.967·29-s + 0.430·31-s + 0.300·32-s + 0.174·33-s + 0.582·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 - 5.60T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 + 9.60T + 61T^{2} \) |
| 67 | \( 1 - 0.605T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 4.69T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 0.697T + 97T^{2} \) |
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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
−7.73016063837010857892890617713, −6.68666249339714856937676872963, −6.18348299834436151824100848999, −5.56407428180727097534750000491, −4.84954266067560095248304939779, −4.20373378970972399297611668126, −3.38768580772079295582688677028, −2.68267978458500747751984847731, −1.24983912169052037195039152677, 0,
1.24983912169052037195039152677, 2.68267978458500747751984847731, 3.38768580772079295582688677028, 4.20373378970972399297611668126, 4.84954266067560095248304939779, 5.56407428180727097534750000491, 6.18348299834436151824100848999, 6.68666249339714856937676872963, 7.73016063837010857892890617713