| L(s) = 1 | − 5-s − 11-s + 13-s − 17-s + 5·19-s − 23-s − 4·25-s + 2·29-s − 6·31-s + 8·37-s + 5·41-s + 43-s + 2·47-s + 6·53-s + 55-s + 2·59-s − 8·61-s − 65-s + 4·67-s − 12·71-s − 14·73-s + 10·79-s + 6·83-s + 85-s − 12·89-s − 5·95-s + 12·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 0.277·13-s − 0.242·17-s + 1.14·19-s − 0.208·23-s − 4/5·25-s + 0.371·29-s − 1.07·31-s + 1.31·37-s + 0.780·41-s + 0.152·43-s + 0.291·47-s + 0.824·53-s + 0.134·55-s + 0.260·59-s − 1.02·61-s − 0.124·65-s + 0.488·67-s − 1.42·71-s − 1.63·73-s + 1.12·79-s + 0.658·83-s + 0.108·85-s − 1.27·89-s − 0.512·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.73096381796228, −13.33533268427699, −12.91379601424149, −12.28704472557767, −11.80272729815818, −11.51500859688668, −10.85622630250160, −10.55165259114821, −9.830843563979278, −9.426488710743947, −8.969023057923592, −8.319369748856487, −7.847492944959426, −7.381942713256978, −7.046057074362057, −6.116019409297014, −5.867999323219160, −5.252542125365308, −4.585971746674632, −4.075925590948762, −3.556364819276286, −2.892575787398110, −2.341906697660920, −1.530302744182015, −0.8313510921772864, 0,
0.8313510921772864, 1.530302744182015, 2.341906697660920, 2.892575787398110, 3.556364819276286, 4.075925590948762, 4.585971746674632, 5.252542125365308, 5.867999323219160, 6.116019409297014, 7.046057074362057, 7.381942713256978, 7.847492944959426, 8.319369748856487, 8.969023057923592, 9.426488710743947, 9.830843563979278, 10.55165259114821, 10.85622630250160, 11.51500859688668, 11.80272729815818, 12.28704472557767, 12.91379601424149, 13.33533268427699, 13.73096381796228
