| L(s) = 1 | + 7-s + 11-s + 13-s − 2·17-s + 5·19-s − 6·23-s + 2·29-s − 3·31-s − 10·37-s + 6·41-s − 3·43-s + 4·47-s − 6·49-s + 10·53-s + 4·59-s + 5·61-s − 5·67-s − 14·73-s + 77-s − 4·79-s − 6·83-s + 14·89-s + 91-s + 15·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.301·11-s + 0.277·13-s − 0.485·17-s + 1.14·19-s − 1.25·23-s + 0.371·29-s − 0.538·31-s − 1.64·37-s + 0.937·41-s − 0.457·43-s + 0.583·47-s − 6/7·49-s + 1.37·53-s + 0.520·59-s + 0.640·61-s − 0.610·67-s − 1.63·73-s + 0.113·77-s − 0.450·79-s − 0.658·83-s + 1.48·89-s + 0.104·91-s + 1.52·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−14.87085330173202, −14.60773485496310, −13.96023961359558, −13.59067830459815, −13.08070271337881, −12.31916477258734, −11.87618509282034, −11.53757530775219, −10.84681138842381, −10.31995197125265, −9.836207188896671, −9.165974089343368, −8.668962409633504, −8.181952568401251, −7.449793316798374, −7.081978485319274, −6.347256149997385, −5.731786911294815, −5.241422965219315, −4.514322548118661, −3.896079068743830, −3.341578774525225, −2.483534788269692, −1.779291204415944, −1.070178437422106, 0,
1.070178437422106, 1.779291204415944, 2.483534788269692, 3.341578774525225, 3.896079068743830, 4.514322548118661, 5.241422965219315, 5.731786911294815, 6.347256149997385, 7.081978485319274, 7.449793316798374, 8.181952568401251, 8.668962409633504, 9.165974089343368, 9.836207188896671, 10.31995197125265, 10.84681138842381, 11.53757530775219, 11.87618509282034, 12.31916477258734, 13.08070271337881, 13.59067830459815, 13.96023961359558, 14.60773485496310, 14.87085330173202
