| L(s) = 1 | + 19·3-s − 19·5-s + 118·9-s − 559·11-s − 282·13-s − 361·15-s − 1.25e3·17-s + 1.95e3·19-s − 2.97e3·23-s − 2.76e3·25-s − 2.37e3·27-s − 62·29-s − 2.03e3·31-s − 1.06e4·33-s + 6.02e3·37-s − 5.35e3·39-s + 2.17e3·41-s + 2.31e4·43-s − 2.24e3·45-s − 2.62e4·47-s − 2.39e4·51-s + 3.02e4·53-s + 1.06e4·55-s + 3.71e4·57-s − 4.49e4·59-s − 2.76e4·61-s + 5.35e3·65-s + ⋯ |
| L(s) = 1 | + 1.21·3-s − 0.339·5-s + 0.485·9-s − 1.39·11-s − 0.462·13-s − 0.414·15-s − 1.05·17-s + 1.24·19-s − 1.17·23-s − 0.884·25-s − 0.626·27-s − 0.0136·29-s − 0.380·31-s − 1.69·33-s + 0.723·37-s − 0.564·39-s + 0.202·41-s + 1.91·43-s − 0.165·45-s − 1.73·47-s − 1.28·51-s + 1.48·53-s + 0.473·55-s + 1.51·57-s − 1.68·59-s − 0.951·61-s + 0.157·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 19 T + p^{5} T^{2} \) |
| 5 | \( 1 + 19 T + p^{5} T^{2} \) |
| 11 | \( 1 + 559 T + p^{5} T^{2} \) |
| 13 | \( 1 + 282 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1259 T + p^{5} T^{2} \) |
| 19 | \( 1 - 103 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 2977 T + p^{5} T^{2} \) |
| 29 | \( 1 + 62 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2037 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6023 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2178 T + p^{5} T^{2} \) |
| 43 | \( 1 - 23180 T + p^{5} T^{2} \) |
| 47 | \( 1 + 26235 T + p^{5} T^{2} \) |
| 53 | \( 1 - 30267 T + p^{5} T^{2} \) |
| 59 | \( 1 + 44965 T + p^{5} T^{2} \) |
| 61 | \( 1 + 27639 T + p^{5} T^{2} \) |
| 67 | \( 1 + 58667 T + p^{5} T^{2} \) |
| 71 | \( 1 + 9520 T + p^{5} T^{2} \) |
| 73 | \( 1 - 6785 T + p^{5} T^{2} \) |
| 79 | \( 1 + 16929 T + p^{5} T^{2} \) |
| 83 | \( 1 - 59572 T + p^{5} T^{2} \) |
| 89 | \( 1 - 51873 T + p^{5} T^{2} \) |
| 97 | \( 1 + 134110 T + p^{5} T^{2} \) |
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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
−11.08533166101565359769298762595, −9.932616478515652755999930660611, −9.075429219367798480553713830677, −7.927393287052803720096766319638, −7.51643405196168242899055294119, −5.78027410751181599638270063862, −4.38636738318164764210876234945, −3.08876708899229142460158175460, −2.13451730939695001953089752105, 0,
2.13451730939695001953089752105, 3.08876708899229142460158175460, 4.38636738318164764210876234945, 5.78027410751181599638270063862, 7.51643405196168242899055294119, 7.927393287052803720096766319638, 9.075429219367798480553713830677, 9.932616478515652755999930660611, 11.08533166101565359769298762595
