| L(s) = 1 | − 9·3-s + 34·5-s + 81·9-s − 332·11-s + 1.02e3·13-s − 306·15-s − 922·17-s − 452·19-s − 3.77e3·23-s − 1.96e3·25-s − 729·27-s + 1.16e3·29-s + 9.79e3·31-s + 2.98e3·33-s + 2.39e3·37-s − 9.23e3·39-s + 7.23e3·41-s + 4.65e3·43-s + 2.75e3·45-s − 2.46e4·47-s + 8.29e3·51-s + 1.11e3·53-s − 1.12e4·55-s + 4.06e3·57-s − 4.68e4·59-s + 9.76e3·61-s + 3.48e4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.608·5-s + 1/3·9-s − 0.827·11-s + 1.68·13-s − 0.351·15-s − 0.773·17-s − 0.287·19-s − 1.48·23-s − 0.630·25-s − 0.192·27-s + 0.257·29-s + 1.83·31-s + 0.477·33-s + 0.287·37-s − 0.972·39-s + 0.671·41-s + 0.383·43-s + 0.202·45-s − 1.62·47-s + 0.446·51-s + 0.0542·53-s − 0.503·55-s + 0.165·57-s − 1.75·59-s + 0.335·61-s + 1.02·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 34 T + p^{5} T^{2} \) |
| 11 | \( 1 + 332 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1026 T + p^{5} T^{2} \) |
| 17 | \( 1 + 922 T + p^{5} T^{2} \) |
| 19 | \( 1 + 452 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3776 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1166 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9792 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2390 T + p^{5} T^{2} \) |
| 41 | \( 1 - 7230 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4652 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24672 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1110 T + p^{5} T^{2} \) |
| 59 | \( 1 + 46892 T + p^{5} T^{2} \) |
| 61 | \( 1 - 9762 T + p^{5} T^{2} \) |
| 67 | \( 1 + 26252 T + p^{5} T^{2} \) |
| 71 | \( 1 - 65440 T + p^{5} T^{2} \) |
| 73 | \( 1 - 5606 T + p^{5} T^{2} \) |
| 79 | \( 1 + 9840 T + p^{5} T^{2} \) |
| 83 | \( 1 + 61108 T + p^{5} T^{2} \) |
| 89 | \( 1 - 62958 T + p^{5} T^{2} \) |
| 97 | \( 1 - 37838 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.657354100114722202553970356612, −8.526089349988505364416914856338, −7.80925921821835176361460696703, −6.23941364260332516456079401392, −6.17492297730751592398327003769, −4.88346513479350880461127393711, −3.87671114838761962616602096941, −2.45757359299401721087276887524, −1.32760186561171369487492882326, 0,
1.32760186561171369487492882326, 2.45757359299401721087276887524, 3.87671114838761962616602096941, 4.88346513479350880461127393711, 6.17492297730751592398327003769, 6.23941364260332516456079401392, 7.80925921821835176361460696703, 8.526089349988505364416914856338, 9.657354100114722202553970356612
