| L(s) = 1 | − 3-s − 5·7-s + 9-s + 5·11-s + 13-s + 3·17-s − 4·19-s + 5·21-s + 5·23-s − 27-s + 4·29-s − 5·33-s − 7·37-s − 39-s + 11·41-s + 12·43-s + 6·47-s + 18·49-s − 3·51-s + 53-s + 4·57-s + 12·59-s + 7·61-s − 5·63-s − 4·67-s − 5·69-s + 7·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 0.727·17-s − 0.917·19-s + 1.09·21-s + 1.04·23-s − 0.192·27-s + 0.742·29-s − 0.870·33-s − 1.15·37-s − 0.160·39-s + 1.71·41-s + 1.82·43-s + 0.875·47-s + 18/7·49-s − 0.420·51-s + 0.137·53-s + 0.529·57-s + 1.56·59-s + 0.896·61-s − 0.629·63-s − 0.488·67-s − 0.601·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.165922489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.165922489\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.19453840117362, −13.79568362806064, −13.08928339761502, −12.60681513461843, −12.39932906321461, −11.88723877653561, −11.18198146997449, −10.68376479422301, −10.24132249127695, −9.579594392102879, −9.226517175547931, −8.852597233044910, −8.092555960805338, −7.138642365092528, −6.906808720358745, −6.424593914159878, −5.898108445944237, −5.502073513170911, −4.527959615327832, −3.902049336228371, −3.604293663842954, −2.821772957401210, −2.136643064274265, −0.9105347020433223, −0.7083384224742470,
0.7083384224742470, 0.9105347020433223, 2.136643064274265, 2.821772957401210, 3.604293663842954, 3.902049336228371, 4.527959615327832, 5.502073513170911, 5.898108445944237, 6.424593914159878, 6.906808720358745, 7.138642365092528, 8.092555960805338, 8.852597233044910, 9.226517175547931, 9.579594392102879, 10.24132249127695, 10.68376479422301, 11.18198146997449, 11.88723877653561, 12.39932906321461, 12.60681513461843, 13.08928339761502, 13.79568362806064, 14.19453840117362
