| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s − 2·7-s + 3·9-s − 2·11-s + 2·12-s − 4·14-s + 16-s + 2·17-s + 6·18-s + 10·19-s − 4·21-s − 4·22-s + 2·23-s + 4·27-s − 2·28-s + 8·29-s − 2·32-s − 4·33-s + 4·34-s + 3·36-s − 6·37-s + 20·38-s − 2·41-s − 8·42-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s − 0.755·7-s + 9-s − 0.603·11-s + 0.577·12-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 1.41·18-s + 2.29·19-s − 0.872·21-s − 0.852·22-s + 0.417·23-s + 0.769·27-s − 0.377·28-s + 1.48·29-s − 0.353·32-s − 0.696·33-s + 0.685·34-s + 1/2·36-s − 0.986·37-s + 3.24·38-s − 0.312·41-s − 1.23·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.916085063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.916085063\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 75 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 159 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 195 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.30861049880386603727007626313, −10.02986799077145690099854831174, −9.551028700846711830484975156993, −9.338464142870070003179233561926, −8.623899970921189425669696093193, −8.306781568806163074528089695870, −7.920672914394022696166674168407, −7.35575436694722070649985946399, −6.88803977241101223317384438115, −6.70722475040716590383159439502, −5.85275843777679050169134605101, −5.32273293730185123531260186709, −5.04420017647999150329751966441, −4.71856178822737784883137397345, −3.73124681105883253137268752324, −3.69008801389471729271519660072, −3.07411770964641053201705364399, −2.79799544958700483458991119345, −1.88711440568386880809605854458, −0.958470111477446266123633130071,
0.958470111477446266123633130071, 1.88711440568386880809605854458, 2.79799544958700483458991119345, 3.07411770964641053201705364399, 3.69008801389471729271519660072, 3.73124681105883253137268752324, 4.71856178822737784883137397345, 5.04420017647999150329751966441, 5.32273293730185123531260186709, 5.85275843777679050169134605101, 6.70722475040716590383159439502, 6.88803977241101223317384438115, 7.35575436694722070649985946399, 7.920672914394022696166674168407, 8.306781568806163074528089695870, 8.623899970921189425669696093193, 9.338464142870070003179233561926, 9.551028700846711830484975156993, 10.02986799077145690099854831174, 10.30861049880386603727007626313
