| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s + 4·8-s + 3·9-s − 4·10-s + 4·11-s − 6·12-s + 4·15-s + 5·16-s + 6·18-s − 6·20-s + 8·22-s + 4·23-s − 8·24-s + 3·25-s − 4·27-s + 8·29-s + 8·30-s + 4·31-s + 6·32-s − 8·33-s + 9·36-s − 8·40-s + 12·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s + 1.20·11-s − 1.73·12-s + 1.03·15-s + 5/4·16-s + 1.41·18-s − 1.34·20-s + 1.70·22-s + 0.834·23-s − 1.63·24-s + 3/5·25-s − 0.769·27-s + 1.48·29-s + 1.46·30-s + 0.718·31-s + 1.06·32-s − 1.39·33-s + 3/2·36-s − 1.26·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.491734200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.491734200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 120 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.637357838871249150992969140618, −9.567674130604693015699260919301, −8.695112981541856694830483106055, −8.627325750954760243245051288257, −7.75150497662837750195125132781, −7.62654815524372747683610256393, −6.98116416165457928505328389820, −6.80802711672635590355621544971, −6.25294328633980436113482368904, −6.16046326002324043497317493969, −5.33334734654136129989780436170, −5.26810567306029973205095177276, −4.52599569155958676488174233941, −4.34758789348115847590335941070, −3.78354034536825266062805513151, −3.62962041399686288396193974671, −2.54437488142531565296789849657, −2.48118261152276074185602676937, −1.02290488520011738099680178258, −0.994829786736176915688651263258,
0.994829786736176915688651263258, 1.02290488520011738099680178258, 2.48118261152276074185602676937, 2.54437488142531565296789849657, 3.62962041399686288396193974671, 3.78354034536825266062805513151, 4.34758789348115847590335941070, 4.52599569155958676488174233941, 5.26810567306029973205095177276, 5.33334734654136129989780436170, 6.16046326002324043497317493969, 6.25294328633980436113482368904, 6.80802711672635590355621544971, 6.98116416165457928505328389820, 7.62654815524372747683610256393, 7.75150497662837750195125132781, 8.627325750954760243245051288257, 8.695112981541856694830483106055, 9.567674130604693015699260919301, 9.637357838871249150992969140618
