| L(s) = 1 | − 5-s + 7-s + 3·11-s − 5·13-s + 8·23-s − 4·25-s − 3·29-s + 3·31-s − 35-s − 10·37-s + 2·41-s + 10·43-s − 4·47-s − 6·49-s − 6·53-s − 3·55-s − 6·59-s − 6·61-s + 5·65-s + 5·67-s − 3·71-s − 2·73-s + 3·77-s − 3·79-s + 17·83-s + 9·89-s − 5·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.904·11-s − 1.38·13-s + 1.66·23-s − 4/5·25-s − 0.557·29-s + 0.538·31-s − 0.169·35-s − 1.64·37-s + 0.312·41-s + 1.52·43-s − 0.583·47-s − 6/7·49-s − 0.824·53-s − 0.404·55-s − 0.781·59-s − 0.768·61-s + 0.620·65-s + 0.610·67-s − 0.356·71-s − 0.234·73-s + 0.341·77-s − 0.337·79-s + 1.86·83-s + 0.953·89-s − 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 139 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.98590427000071, −16.45817357854213, −15.63375579033415, −15.23473900488396, −14.55872065044519, −14.27455716919986, −13.50728389175900, −12.78403220434020, −12.17208608738889, −11.82562376298521, −11.07910578491263, −10.65090462667448, −9.681353296682247, −9.321758739824451, −8.660019576581538, −7.825448870403683, −7.372413500863761, −6.761016940695273, −6.000795444130883, −5.040821724506133, −4.680951576504720, −3.794185831964092, −3.073428865401004, −2.138727075477678, −1.221141727208652, 0,
1.221141727208652, 2.138727075477678, 3.073428865401004, 3.794185831964092, 4.680951576504720, 5.040821724506133, 6.000795444130883, 6.761016940695273, 7.372413500863761, 7.825448870403683, 8.660019576581538, 9.321758739824451, 9.681353296682247, 10.65090462667448, 11.07910578491263, 11.82562376298521, 12.17208608738889, 12.78403220434020, 13.50728389175900, 14.27455716919986, 14.55872065044519, 15.23473900488396, 15.63375579033415, 16.45817357854213, 16.98590427000071
