| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 2·5-s + 4·6-s − 9-s + 4·10-s − 2·11-s + 2·12-s + 4·15-s + 16-s − 4·17-s − 2·18-s − 12·19-s + 2·20-s − 4·22-s − 4·23-s − 7·25-s − 6·27-s + 2·29-s + 8·30-s − 6·31-s − 2·32-s − 4·33-s − 8·34-s − 36-s + 8·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 1/3·9-s + 1.26·10-s − 0.603·11-s + 0.577·12-s + 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.471·18-s − 2.75·19-s + 0.447·20-s − 0.852·22-s − 0.834·23-s − 7/5·25-s − 1.15·27-s + 0.371·29-s + 1.46·30-s − 1.07·31-s − 0.353·32-s − 0.696·33-s − 1.37·34-s − 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24019801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24019801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098812055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098812055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 26 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 77 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.314720156160606557338415497873, −8.102527743025591961774710065072, −7.84035950804441158807286804040, −7.49217634401178810892653193724, −6.74748947982473720139617500563, −6.46008836667705081090101529741, −6.26334923064031677823555086777, −5.80330650749470743352141621767, −5.39035887429268365234004528706, −5.27986704764013123292418839256, −4.57667377401678358266661729555, −4.23676352385138068008824231628, −3.96971455833825525451249231232, −3.81805161478678847959605216015, −2.92521073980771792294134883656, −2.81113874098421763541657296480, −2.25914505468038502761622854291, −1.93930744292217871525707337430, −1.65391913647347312471805383466, −0.17484454551274335413546290379,
0.17484454551274335413546290379, 1.65391913647347312471805383466, 1.93930744292217871525707337430, 2.25914505468038502761622854291, 2.81113874098421763541657296480, 2.92521073980771792294134883656, 3.81805161478678847959605216015, 3.96971455833825525451249231232, 4.23676352385138068008824231628, 4.57667377401678358266661729555, 5.27986704764013123292418839256, 5.39035887429268365234004528706, 5.80330650749470743352141621767, 6.26334923064031677823555086777, 6.46008836667705081090101529741, 6.74748947982473720139617500563, 7.49217634401178810892653193724, 7.84035950804441158807286804040, 8.102527743025591961774710065072, 8.314720156160606557338415497873
