| L(s) = 1 | − 4·2-s − 3-s + 12·4-s − 10·5-s + 4·6-s − 6·7-s − 32·8-s + 39·9-s + 40·10-s − 72·11-s − 12·12-s − 111·13-s + 24·14-s + 10·15-s + 80·16-s − 124·17-s − 156·18-s − 22·19-s − 120·20-s + 6·21-s + 288·22-s + 32·24-s − 134·25-s + 444·26-s − 104·27-s − 72·28-s + 15·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.894·5-s + 0.272·6-s − 0.323·7-s − 1.41·8-s + 13/9·9-s + 1.26·10-s − 1.97·11-s − 0.288·12-s − 2.36·13-s + 0.458·14-s + 0.172·15-s + 5/4·16-s − 1.76·17-s − 2.04·18-s − 0.265·19-s − 1.34·20-s + 0.0623·21-s + 2.79·22-s + 0.272·24-s − 1.07·25-s + 3.34·26-s − 0.741·27-s − 0.485·28-s + 0.0960·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T - 38 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 234 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 654 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 72 T + 3302 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 111 T + 6972 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 7766 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 22 T + 10518 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 5528 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 67 T + 57742 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 18 T + 51162 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 485 T + 192128 T^{2} - 485 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 440 T + 175270 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 215 T + 130550 T^{2} - 215 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 240 T + 154550 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 792 T + 9442 p T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 456 T + 45270 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 240 T + 592310 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 705 T + 755198 T^{2} + 705 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 27 T + 769596 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 594 T + 1018158 T^{2} + 594 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 394 T + 857622 T^{2} + 394 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 486 T + 1378418 T^{2} - 486 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2152 T + 2707438 T^{2} + 2152 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.274916036245247292575034671837, −9.188527307320391161685325274140, −8.192275769896006152898760493803, −8.173213749286640820639464242973, −7.53003937772791443401615322303, −7.37295168625212327313510772706, −7.19061128706771043891229551372, −6.65835127134340565312980835274, −5.93048726626522193714140732764, −5.65858945250457001074064441450, −4.68759863406747565953411361114, −4.61771644498010179329481817230, −4.16247731222779209801379471828, −3.27545767405754648387193491196, −2.52554858898111036495061197344, −2.35606099798607846028301590900, −1.81616623609537440203383926863, −0.70923627208182498051446656368, 0, 0,
0.70923627208182498051446656368, 1.81616623609537440203383926863, 2.35606099798607846028301590900, 2.52554858898111036495061197344, 3.27545767405754648387193491196, 4.16247731222779209801379471828, 4.61771644498010179329481817230, 4.68759863406747565953411361114, 5.65858945250457001074064441450, 5.93048726626522193714140732764, 6.65835127134340565312980835274, 7.19061128706771043891229551372, 7.37295168625212327313510772706, 7.53003937772791443401615322303, 8.173213749286640820639464242973, 8.192275769896006152898760493803, 9.188527307320391161685325274140, 9.274916036245247292575034671837
