| L(s) = 1 | − 2-s + 2.39·3-s + 4-s + 4.06·5-s − 2.39·6-s − 3.19·7-s − 8-s + 2.75·9-s − 4.06·10-s − 5.11·11-s + 2.39·12-s + 13-s + 3.19·14-s + 9.74·15-s + 16-s − 5.04·17-s − 2.75·18-s + 4.06·20-s − 7.67·21-s + 5.11·22-s − 0.882·23-s − 2.39·24-s + 11.4·25-s − 26-s − 0.592·27-s − 3.19·28-s − 5.65·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.38·3-s + 0.5·4-s + 1.81·5-s − 0.979·6-s − 1.20·7-s − 0.353·8-s + 0.917·9-s − 1.28·10-s − 1.54·11-s + 0.692·12-s + 0.277·13-s + 0.855·14-s + 2.51·15-s + 0.250·16-s − 1.22·17-s − 0.648·18-s + 0.908·20-s − 1.67·21-s + 1.08·22-s − 0.183·23-s − 0.489·24-s + 2.29·25-s − 0.196·26-s − 0.114·27-s − 0.604·28-s − 1.04·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 23 | \( 1 + 0.882T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 9.37T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 - 0.205T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 - 5.49T + 59T^{2} \) |
| 61 | \( 1 + 0.172T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 2.40T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 6.06T + 83T^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−7.41132793490147649036897755751, −6.79826944231036777125768562265, −6.08835084228027606351291070138, −5.58176068571426343394555508974, −4.56014700951268727878879480232, −3.33414967641568260431997518707, −2.72199031993888373917091770082, −2.33732461332811544237824038536, −1.55939006147672843664745150325, 0,
1.55939006147672843664745150325, 2.33732461332811544237824038536, 2.72199031993888373917091770082, 3.33414967641568260431997518707, 4.56014700951268727878879480232, 5.58176068571426343394555508974, 6.08835084228027606351291070138, 6.79826944231036777125768562265, 7.41132793490147649036897755751
