| L(s) = 1 | − 2·2-s + 4-s + 4·11-s + 2·13-s + 16-s + 4·17-s − 8·22-s − 8·23-s − 4·26-s − 4·29-s − 8·31-s + 2·32-s − 8·34-s + 4·37-s − 16·41-s − 8·43-s + 4·44-s + 16·46-s − 12·47-s − 6·49-s + 2·52-s − 4·53-s + 8·58-s − 4·59-s + 4·61-s + 16·62-s − 11·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 1.70·22-s − 1.66·23-s − 0.784·26-s − 0.742·29-s − 1.43·31-s + 0.353·32-s − 1.37·34-s + 0.657·37-s − 2.49·41-s − 1.21·43-s + 0.603·44-s + 2.35·46-s − 1.75·47-s − 6/7·49-s + 0.277·52-s − 0.549·53-s + 1.05·58-s − 0.520·59-s + 0.512·61-s + 2.03·62-s − 1.37·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.505388825145810989734973295844, −8.376248915175191012555886378761, −7.85553932478053997118806309679, −7.78125857915822251154855089799, −7.12706808297999904234475406061, −6.66736800084159976178806314218, −6.48945160884180651263991581523, −6.00728168354301094205070179107, −5.44833412210563797158897097256, −5.37684928804853599847359036333, −4.46636512722584703438379674054, −4.29190556091377468742489018029, −3.63399792816148471227542783963, −3.32941242127549214993476156880, −2.95990805647709554937300916170, −1.92941528087342088140008734365, −1.50207129878251267066730011727, −1.34994662759355705531577552425, 0, 0,
1.34994662759355705531577552425, 1.50207129878251267066730011727, 1.92941528087342088140008734365, 2.95990805647709554937300916170, 3.32941242127549214993476156880, 3.63399792816148471227542783963, 4.29190556091377468742489018029, 4.46636512722584703438379674054, 5.37684928804853599847359036333, 5.44833412210563797158897097256, 6.00728168354301094205070179107, 6.48945160884180651263991581523, 6.66736800084159976178806314218, 7.12706808297999904234475406061, 7.78125857915822251154855089799, 7.85553932478053997118806309679, 8.376248915175191012555886378761, 8.505388825145810989734973295844
