| L(s) = 1 | + 3-s − 2·5-s + 9-s + 2·13-s − 2·15-s − 6·17-s + 4·23-s − 25-s + 27-s + 2·29-s + 10·37-s + 2·39-s − 6·41-s − 8·43-s − 2·45-s − 4·47-s − 7·49-s − 6·51-s + 6·53-s + 12·59-s + 2·61-s − 4·65-s − 4·67-s + 4·69-s + 12·71-s + 14·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s − 0.583·47-s − 49-s − 0.840·51-s + 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s + 0.481·69-s + 1.42·71-s + 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.53270579865647, −15.24768637007119, −14.83081112783990, −14.13053576840583, −13.47242722446640, −13.12111009976164, −12.64460193257121, −11.79629959187446, −11.29038786188097, −11.09597723234223, −10.13886761664453, −9.698609780125681, −8.934982332853158, −8.419938134275838, −8.131602229665511, −7.336452758064115, −6.761343226780467, −6.324268278821378, −5.309085738805941, −4.657198551721381, −4.054546947387842, −3.512093564653108, −2.766037529810022, −2.030044641029576, −1.058438676786885, 0,
1.058438676786885, 2.030044641029576, 2.766037529810022, 3.512093564653108, 4.054546947387842, 4.657198551721381, 5.309085738805941, 6.324268278821378, 6.761343226780467, 7.336452758064115, 8.131602229665511, 8.419938134275838, 8.934982332853158, 9.698609780125681, 10.13886761664453, 11.09597723234223, 11.29038786188097, 11.79629959187446, 12.64460193257121, 13.12111009976164, 13.47242722446640, 14.13053576840583, 14.83081112783990, 15.24768637007119, 15.53270579865647
