| L(s) = 1 | − 3-s + 9-s − 4·11-s − 13-s + 6·17-s + 8·19-s + 8·23-s − 27-s + 2·29-s − 8·31-s + 4·33-s + 10·37-s + 39-s + 6·41-s − 4·43-s − 7·49-s − 6·51-s + 14·53-s − 8·57-s + 12·59-s − 10·61-s − 8·67-s − 8·69-s + 14·73-s + 4·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 1.45·17-s + 1.83·19-s + 1.66·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 1.64·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 49-s − 0.840·51-s + 1.92·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.977·67-s − 0.963·69-s + 1.63·73-s + 0.450·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.102370717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.102370717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.12001566310034, −14.59632203376995, −14.08824212477434, −13.25667602799703, −13.08201858295690, −12.39659579019613, −11.93135672704122, −11.30314665956625, −10.89716953951294, −10.29021462387686, −9.646209814269608, −9.438432050751252, −8.498155936386813, −7.825112800732630, −7.356931134143752, −7.054663332627447, −5.996521569647491, −5.506627138009289, −5.152612396764761, −4.558618585382864, −3.496777859057040, −3.075595966202268, −2.317666178639684, −1.185015566033281, −0.6647780178570534,
0.6647780178570534, 1.185015566033281, 2.317666178639684, 3.075595966202268, 3.496777859057040, 4.558618585382864, 5.152612396764761, 5.506627138009289, 5.996521569647491, 7.054663332627447, 7.356931134143752, 7.825112800732630, 8.498155936386813, 9.438432050751252, 9.646209814269608, 10.29021462387686, 10.89716953951294, 11.30314665956625, 11.93135672704122, 12.39659579019613, 13.08201858295690, 13.25667602799703, 14.08824212477434, 14.59632203376995, 15.12001566310034
