| L(s) = 1 | − 3-s + 1.65·5-s − 0.854·7-s + 9-s − 2.53·11-s + 4.20·13-s − 1.65·15-s − 2.20·17-s + 0.531·19-s + 0.854·21-s + 2.53·23-s − 2.27·25-s − 27-s + 1.88·29-s − 5.67·31-s + 2.53·33-s − 1.41·35-s + 5.08·37-s − 4.20·39-s − 2.98·41-s − 2.40·43-s + 1.65·45-s − 5.56·47-s − 6.26·49-s + 2.20·51-s − 9.77·53-s − 4.17·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.738·5-s − 0.322·7-s + 0.333·9-s − 0.763·11-s + 1.16·13-s − 0.426·15-s − 0.535·17-s + 0.121·19-s + 0.186·21-s + 0.527·23-s − 0.454·25-s − 0.192·27-s + 0.350·29-s − 1.01·31-s + 0.440·33-s − 0.238·35-s + 0.836·37-s − 0.674·39-s − 0.465·41-s − 0.366·43-s + 0.246·45-s − 0.811·47-s − 0.895·49-s + 0.309·51-s − 1.34·53-s − 0.563·55-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 | \( 1 - 1.65T + 5T^{2} \) |
| 7 | \( 1 + 0.854T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 4.20T + 13T^{2} \) |
| 17 | \( 1 + 2.20T + 17T^{2} \) |
| 19 | \( 1 - 0.531T + 19T^{2} \) |
| 23 | \( 1 - 2.53T + 23T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 - 5.08T + 37T^{2} \) |
| 41 | \( 1 + 2.98T + 41T^{2} \) |
| 43 | \( 1 + 2.40T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + 9.77T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 4.71T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 + 2.89T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 4.98T + 79T^{2} \) |
| 83 | \( 1 - 8.44T + 83T^{2} \) |
| 89 | \( 1 - 7.85T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.46022236366706553557771908492, −6.44542375743025286391742864213, −6.28413857199537001364115837126, −5.39414431411289418965659646533, −4.89754354572355608462360623172, −3.89481545363778758319792700499, −3.11306508177960060017045051779, −2.12818723407021074278669266929, −1.26954198384440586475909508206, 0,
1.26954198384440586475909508206, 2.12818723407021074278669266929, 3.11306508177960060017045051779, 3.89481545363778758319792700499, 4.89754354572355608462360623172, 5.39414431411289418965659646533, 6.28413857199537001364115837126, 6.44542375743025286391742864213, 7.46022236366706553557771908492
