| L(s) = 1 | − 2·3-s − 2·4-s − 4·5-s + 3·9-s + 2·11-s + 4·12-s − 4·13-s + 8·15-s − 2·17-s + 8·19-s + 8·20-s + 4·25-s − 4·27-s + 2·29-s + 6·31-s − 4·33-s − 6·36-s − 2·37-s + 8·39-s + 2·41-s − 10·43-s − 4·44-s − 12·45-s − 18·47-s + 4·51-s + 8·52-s + 8·53-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 1.78·5-s + 9-s + 0.603·11-s + 1.15·12-s − 1.10·13-s + 2.06·15-s − 0.485·17-s + 1.83·19-s + 1.78·20-s + 4/5·25-s − 0.769·27-s + 0.371·29-s + 1.07·31-s − 0.696·33-s − 36-s − 0.328·37-s + 1.28·39-s + 0.312·41-s − 1.52·43-s − 0.603·44-s − 1.78·45-s − 2.62·47-s + 0.560·51-s + 1.10·52-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 173 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 91 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81536128051432299388540779343, −7.68542676738369525551702232502, −7.09605216694609755127372990738, −6.84727915170172598621879760082, −6.62297992810949274203209241665, −6.23042952138944208166638268726, −5.50118210440411029203220875502, −5.32623041096835230558497133466, −4.90542458372635834329249302918, −4.70513986997764905516749695502, −4.29492636100223364127288075987, −4.03332972064782946531072227623, −3.41170808101923195852329295083, −3.38508879765128773443871102874, −2.65769814446075140339979762284, −2.00559952499735569126155279467, −1.26491025786676716303416602889, −0.819393996952786040339090110820, 0, 0,
0.819393996952786040339090110820, 1.26491025786676716303416602889, 2.00559952499735569126155279467, 2.65769814446075140339979762284, 3.38508879765128773443871102874, 3.41170808101923195852329295083, 4.03332972064782946531072227623, 4.29492636100223364127288075987, 4.70513986997764905516749695502, 4.90542458372635834329249302918, 5.32623041096835230558497133466, 5.50118210440411029203220875502, 6.23042952138944208166638268726, 6.62297992810949274203209241665, 6.84727915170172598621879760082, 7.09605216694609755127372990738, 7.68542676738369525551702232502, 7.81536128051432299388540779343
