| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s − 4·11-s + 5·16-s + 6·20-s + 8·22-s − 4·23-s + 3·25-s − 8·29-s + 4·31-s − 6·32-s − 8·40-s − 12·41-s + 12·43-s − 12·44-s + 8·46-s − 4·47-s − 6·50-s − 4·53-s − 8·55-s + 16·58-s − 12·59-s + 12·61-s − 8·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 1.26·10-s − 1.20·11-s + 5/4·16-s + 1.34·20-s + 1.70·22-s − 0.834·23-s + 3/5·25-s − 1.48·29-s + 0.718·31-s − 1.06·32-s − 1.26·40-s − 1.87·41-s + 1.82·43-s − 1.80·44-s + 1.17·46-s − 0.583·47-s − 0.848·50-s − 0.549·53-s − 1.07·55-s + 2.10·58-s − 1.56·59-s + 1.53·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 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
−8.047510392675918253894920098590, −8.024961914113575635418295164110, −7.46014361349206430950087934023, −7.32287935039268121435515831255, −6.73445762050930757214098397557, −6.50142866503632519721682440244, −6.02133968880182180583180009152, −5.68119668602881275579526228020, −5.38276377125478138242689117389, −5.05982179369841229061226997834, −4.31398259262026548774970353272, −4.07872466930935751139230104927, −3.25084160218310597280720462982, −3.00184758640857949842927071778, −2.41366557228148687647954443400, −2.20972809059151474681798338423, −1.41713118953117451274512190717, −1.33630410923241965063319200669, 0, 0,
1.33630410923241965063319200669, 1.41713118953117451274512190717, 2.20972809059151474681798338423, 2.41366557228148687647954443400, 3.00184758640857949842927071778, 3.25084160218310597280720462982, 4.07872466930935751139230104927, 4.31398259262026548774970353272, 5.05982179369841229061226997834, 5.38276377125478138242689117389, 5.68119668602881275579526228020, 6.02133968880182180583180009152, 6.50142866503632519721682440244, 6.73445762050930757214098397557, 7.32287935039268121435515831255, 7.46014361349206430950087934023, 8.024961914113575635418295164110, 8.047510392675918253894920098590
