L(s) = 1 | + 2.20·2-s − 3.00·3-s + 2.86·4-s + 5-s − 6.62·6-s − 7-s + 1.89·8-s + 6.02·9-s + 2.20·10-s + 0.0808·11-s − 8.59·12-s − 4.39·13-s − 2.20·14-s − 3.00·15-s − 1.53·16-s + 2.16·17-s + 13.2·18-s + 4.06·19-s + 2.86·20-s + 3.00·21-s + 0.178·22-s + 6.76·23-s − 5.70·24-s + 25-s − 9.68·26-s − 9.09·27-s − 2.86·28-s + ⋯ |
L(s) = 1 | + 1.55·2-s − 1.73·3-s + 1.43·4-s + 0.447·5-s − 2.70·6-s − 0.377·7-s + 0.670·8-s + 2.00·9-s + 0.697·10-s + 0.0243·11-s − 2.48·12-s − 1.21·13-s − 0.589·14-s − 0.775·15-s − 0.384·16-s + 0.525·17-s + 3.13·18-s + 0.932·19-s + 0.639·20-s + 0.655·21-s + 0.0379·22-s + 1.40·23-s − 1.16·24-s + 0.200·25-s − 1.90·26-s − 1.75·27-s − 0.540·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 3 | \( 1 + 3.00T + 3T^{2} \) |
| 11 | \( 1 - 0.0808T + 11T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 - 2.16T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 - 2.41T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 - 4.33T + 59T^{2} \) |
| 61 | \( 1 + 1.80T + 61T^{2} \) |
| 67 | \( 1 - 0.458T + 67T^{2} \) |
| 71 | \( 1 + 2.72T + 71T^{2} \) |
| 73 | \( 1 - 4.02T + 73T^{2} \) |
| 79 | \( 1 - 1.55T + 79T^{2} \) |
| 83 | \( 1 + 0.347T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 5.13T + 97T^{2} \) |
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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
−7.10115859125498557093804504744, −6.55792188050419580302991952756, −5.72025602905950483760860368123, −5.37929913781832898909200529670, −4.99898965194243216453932398163, −4.17023904546320983941548738757, −3.34413398528054490995116006636, −2.44618487665011507007168659507, −1.29129537727653860122709098357, 0,
1.29129537727653860122709098357, 2.44618487665011507007168659507, 3.34413398528054490995116006636, 4.17023904546320983941548738757, 4.99898965194243216453932398163, 5.37929913781832898909200529670, 5.72025602905950483760860368123, 6.55792188050419580302991952756, 7.10115859125498557093804504744