L(s) = 1 | + 1.28·2-s − 0.356·4-s − 2.16·5-s + 1.04·7-s − 3.02·8-s − 2.78·10-s + 4.47·11-s + 1.34·14-s − 3.15·16-s − 1.67·17-s − 6.13·19-s + 0.774·20-s + 5.74·22-s + 6.78·23-s − 0.295·25-s − 0.374·28-s − 9.80·29-s − 7.91·31-s + 1.99·32-s − 2.14·34-s − 2.27·35-s − 10.1·37-s − 7.86·38-s + 6.55·40-s − 7.32·41-s − 0.506·43-s − 1.59·44-s + ⋯ |
L(s) = 1 | + 0.906·2-s − 0.178·4-s − 0.969·5-s + 0.396·7-s − 1.06·8-s − 0.879·10-s + 1.35·11-s + 0.359·14-s − 0.789·16-s − 0.406·17-s − 1.40·19-s + 0.173·20-s + 1.22·22-s + 1.41·23-s − 0.0591·25-s − 0.0707·28-s − 1.82·29-s − 1.42·31-s + 0.352·32-s − 0.368·34-s − 0.384·35-s − 1.66·37-s − 1.27·38-s + 1.03·40-s − 1.14·41-s − 0.0771·43-s − 0.240·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 + 6.13T + 19T^{2} \) |
| 23 | \( 1 - 6.78T + 23T^{2} \) |
| 29 | \( 1 + 9.80T + 29T^{2} \) |
| 31 | \( 1 + 7.91T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 + 0.506T + 43T^{2} \) |
| 47 | \( 1 + 0.887T + 47T^{2} \) |
| 53 | \( 1 + 5.11T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 - 1.36T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 - 3.87T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 + 4.81T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.823224933012562442036060225907, −8.515229802945749306592386321439, −7.22859199333072206252404204647, −6.63826547301163654180648468968, −5.55018319951028662796986349673, −4.73378026642536410377110283432, −3.87786765902530675949602612546, −3.48950618314885197331659682943, −1.84874921150038515210556530714, 0,
1.84874921150038515210556530714, 3.48950618314885197331659682943, 3.87786765902530675949602612546, 4.73378026642536410377110283432, 5.55018319951028662796986349673, 6.63826547301163654180648468968, 7.22859199333072206252404204647, 8.515229802945749306592386321439, 8.823224933012562442036060225907