| L(s) = 1 | − 6·3-s + 6·5-s − 10·7-s + 27·9-s + 22·11-s + 2·13-s − 36·15-s − 104·17-s + 4·19-s + 60·21-s + 102·23-s − 206·25-s − 108·27-s + 392·29-s + 64·31-s − 132·33-s − 60·35-s + 164·37-s − 12·39-s − 732·41-s + 168·43-s + 162·45-s + 314·47-s − 594·49-s + 624·51-s + 382·53-s + 132·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.536·5-s − 0.539·7-s + 9-s + 0.603·11-s + 0.0426·13-s − 0.619·15-s − 1.48·17-s + 0.0482·19-s + 0.623·21-s + 0.924·23-s − 1.64·25-s − 0.769·27-s + 2.51·29-s + 0.370·31-s − 0.696·33-s − 0.289·35-s + 0.728·37-s − 0.0492·39-s − 2.78·41-s + 0.595·43-s + 0.536·45-s + 0.974·47-s − 1.73·49-s + 1.71·51-s + 0.990·53-s + 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.977147018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977147018\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T + 242 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 694 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 518 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 104 T + 7022 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T - 1578 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 102 T + 24062 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 392 T + 75702 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 64 T + 33406 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 164 T - 770 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 732 T + 264998 T^{2} + 732 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 168 T + 2034 p T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 314 T + 4210 p T^{2} - 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 382 T + 310962 T^{2} - 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 508 T + 418086 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 66354 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 216 T + 298758 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 878 T + 841070 T^{2} - 878 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 260 T - 58058 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 118 T + 829606 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 496 T + 441030 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1756 T + 1701014 T^{2} + 1756 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1968 T + 2769054 T^{2} - 1968 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.021454127616533426357175363647, −8.554681444071089149567639542475, −8.149850296927034707385892733536, −7.894018919936131613829808631499, −6.92937398770585090619961134049, −6.91979985338095484621837850440, −6.51036132277370825999220237496, −6.42234226331580791995352119290, −5.69764915832630193551818137380, −5.52086896829320401916140847523, −4.87827274909746187991101905205, −4.66134121060462046454581958654, −4.10471962081641807545220563195, −3.75241912374555422187080648040, −3.06915114451687480472766461542, −2.56823710672459861207511344822, −1.98654270593494991972017557650, −1.50264529427403396940887726389, −0.77192268720630464833456094757, −0.41862038056931748637911620521,
0.41862038056931748637911620521, 0.77192268720630464833456094757, 1.50264529427403396940887726389, 1.98654270593494991972017557650, 2.56823710672459861207511344822, 3.06915114451687480472766461542, 3.75241912374555422187080648040, 4.10471962081641807545220563195, 4.66134121060462046454581958654, 4.87827274909746187991101905205, 5.52086896829320401916140847523, 5.69764915832630193551818137380, 6.42234226331580791995352119290, 6.51036132277370825999220237496, 6.91979985338095484621837850440, 6.92937398770585090619961134049, 7.894018919936131613829808631499, 8.149850296927034707385892733536, 8.554681444071089149567639542475, 9.021454127616533426357175363647
