| L(s) = 1 | + 2·2-s + 2·3-s − 4-s + 4·6-s − 8·8-s + 3·9-s + 2·11-s − 2·12-s + 9·13-s − 7·16-s + 17-s + 6·18-s − 2·19-s + 4·22-s − 16·24-s + 18·26-s + 4·27-s + 11·29-s + 10·31-s + 14·32-s + 4·33-s + 2·34-s − 3·36-s + 5·37-s − 4·38-s + 18·39-s − 5·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 1/2·4-s + 1.63·6-s − 2.82·8-s + 9-s + 0.603·11-s − 0.577·12-s + 2.49·13-s − 7/4·16-s + 0.242·17-s + 1.41·18-s − 0.458·19-s + 0.852·22-s − 3.26·24-s + 3.53·26-s + 0.769·27-s + 2.04·29-s + 1.79·31-s + 2.47·32-s + 0.696·33-s + 0.342·34-s − 1/2·36-s + 0.821·37-s − 0.648·38-s + 2.88·39-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.085512376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.085512376\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 3 p T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 49 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 87 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 111 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 121 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 219 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 213 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.058052298791271090132899876655, −9.002323657462668260045432006373, −8.717917507663199580099875647359, −8.342916251547541267608811467253, −8.062342299095762193361618769794, −7.67936052246176006397150435575, −6.63962792386534838733971776413, −6.58509042088450208041408405114, −6.05761204268223850290119130945, −6.00576252668532945349096965573, −5.16098011382932892381786557334, −4.78481468581057925511933073923, −4.38901518237591598027508741045, −4.03927744598188054867435992876, −3.62018348495749946622227172830, −3.31996544183404014002537374828, −2.86783374174408485044327812798, −2.32725131854991977661984969121, −1.21606054122932682573547065653, −0.886968208092717484353992052130,
0.886968208092717484353992052130, 1.21606054122932682573547065653, 2.32725131854991977661984969121, 2.86783374174408485044327812798, 3.31996544183404014002537374828, 3.62018348495749946622227172830, 4.03927744598188054867435992876, 4.38901518237591598027508741045, 4.78481468581057925511933073923, 5.16098011382932892381786557334, 6.00576252668532945349096965573, 6.05761204268223850290119130945, 6.58509042088450208041408405114, 6.63962792386534838733971776413, 7.67936052246176006397150435575, 8.062342299095762193361618769794, 8.342916251547541267608811467253, 8.717917507663199580099875647359, 9.002323657462668260045432006373, 9.058052298791271090132899876655
