L(s) = 1 | + 0.938·2-s − 0.687·3-s − 1.11·4-s − 5-s − 0.645·6-s − 7-s − 2.92·8-s − 2.52·9-s − 0.938·10-s − 2.61·11-s + 0.769·12-s + 0.664·13-s − 0.938·14-s + 0.687·15-s − 0.509·16-s + 1.47·17-s − 2.37·18-s + 8.67·19-s + 1.11·20-s + 0.687·21-s − 2.45·22-s − 1.00·23-s + 2.01·24-s + 25-s + 0.623·26-s + 3.80·27-s + 1.11·28-s + ⋯ |
L(s) = 1 | + 0.663·2-s − 0.397·3-s − 0.559·4-s − 0.447·5-s − 0.263·6-s − 0.377·7-s − 1.03·8-s − 0.842·9-s − 0.296·10-s − 0.787·11-s + 0.222·12-s + 0.184·13-s − 0.250·14-s + 0.177·15-s − 0.127·16-s + 0.357·17-s − 0.559·18-s + 1.98·19-s + 0.250·20-s + 0.150·21-s − 0.522·22-s − 0.209·23-s + 0.411·24-s + 0.200·25-s + 0.122·26-s + 0.731·27-s + 0.211·28-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 | \( 1 - 0.938T + 2T^{2} \) |
| 3 | \( 1 + 0.687T + 3T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 - 0.664T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 - 8.67T + 19T^{2} \) |
| 23 | \( 1 + 1.00T + 23T^{2} \) |
| 29 | \( 1 - 6.73T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 + 1.50T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 + 4.17T + 47T^{2} \) |
| 53 | \( 1 + 8.29T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 6.03T + 67T^{2} \) |
| 71 | \( 1 - 5.79T + 71T^{2} \) |
| 73 | \( 1 + 0.203T + 73T^{2} \) |
| 79 | \( 1 + 6.25T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 6.42T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−7.49232303197060222475854566958, −6.61322134459631580569401829919, −5.90294473633719736197507947616, −5.16353137778429101376781391852, −5.00663931307139504161691718383, −3.81382419139261779340200423637, −3.25300830684885667010759398296, −2.65057035120131589808141792628, −0.978250671128987457411758596898, 0,
0.978250671128987457411758596898, 2.65057035120131589808141792628, 3.25300830684885667010759398296, 3.81382419139261779340200423637, 5.00663931307139504161691718383, 5.16353137778429101376781391852, 5.90294473633719736197507947616, 6.61322134459631580569401829919, 7.49232303197060222475854566958