L(s) = 1 | + 3-s − 0.970·5-s − 4.67·7-s + 9-s + 1.74·11-s + 4.10·13-s − 0.970·15-s − 4.80·17-s + 1.30·19-s − 4.67·21-s + 5.16·23-s − 4.05·25-s + 27-s − 4.44·29-s + 1.87·31-s + 1.74·33-s + 4.53·35-s − 3.00·37-s + 4.10·39-s + 5.81·41-s + 4.95·43-s − 0.970·45-s + 2.05·47-s + 14.8·49-s − 4.80·51-s − 8.71·53-s − 1.69·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.433·5-s − 1.76·7-s + 0.333·9-s + 0.525·11-s + 1.13·13-s − 0.250·15-s − 1.16·17-s + 0.300·19-s − 1.01·21-s + 1.07·23-s − 0.811·25-s + 0.192·27-s − 0.825·29-s + 0.336·31-s + 0.303·33-s + 0.766·35-s − 0.493·37-s + 0.657·39-s + 0.907·41-s + 0.755·43-s − 0.144·45-s + 0.299·47-s + 2.11·49-s − 0.673·51-s − 1.19·53-s − 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 0.970T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 23 | \( 1 - 5.16T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 + 3.00T + 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 - 2.05T + 47T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 + 6.51T + 79T^{2} \) |
| 83 | \( 1 - 0.0124T + 83T^{2} \) |
| 89 | \( 1 + 6.84T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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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
−7.36029778287294183439376306672, −6.82883936444405989363525045817, −6.24911952603192263391047789048, −5.57894777321449814878445138603, −4.30399703336587295257452360610, −3.82256824544201217249551489364, −3.19563624630525217676772539702, −2.44986238413409714761071726267, −1.21307718476042420978197725032, 0,
1.21307718476042420978197725032, 2.44986238413409714761071726267, 3.19563624630525217676772539702, 3.82256824544201217249551489364, 4.30399703336587295257452360610, 5.57894777321449814878445138603, 6.24911952603192263391047789048, 6.82883936444405989363525045817, 7.36029778287294183439376306672