| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s − 4·8-s + 2·9-s − 4·10-s + 6·11-s + 6·12-s + 4·13-s + 4·14-s + 4·15-s + 5·16-s + 2·17-s − 4·18-s − 8·19-s + 6·20-s − 4·21-s − 12·22-s + 16·23-s − 8·24-s − 2·25-s − 8·26-s + 6·27-s − 6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s + 1.80·11-s + 1.73·12-s + 1.10·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s − 0.942·18-s − 1.83·19-s + 1.34·20-s − 0.872·21-s − 2.55·22-s + 3.33·23-s − 1.63·24-s − 2/5·25-s − 1.56·26-s + 1.15·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.357273275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.357273275\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−12.28832103067981868195069364999, −11.92612142471837471706435068419, −10.98017512674928301187353752954, −10.90494291763676779852377629713, −10.36006400201378464173297109921, −9.752887239297725819796461544628, −9.173706923637329481742301738429, −9.103938484655653492425357032401, −8.610264838777983408651887538747, −8.451991930124608559593159911555, −7.41093439925521310147741227555, −6.83921001446596737633967757560, −6.66062702753093859465403758477, −6.11347826531456823537239071134, −5.31932687459869695779278148644, −4.23178491420962269647117850282, −3.28061967569918783190091349429, −3.13391409231110049395176238944, −1.87504138844501808137256975525, −1.36349622199492200919804644331,
1.36349622199492200919804644331, 1.87504138844501808137256975525, 3.13391409231110049395176238944, 3.28061967569918783190091349429, 4.23178491420962269647117850282, 5.31932687459869695779278148644, 6.11347826531456823537239071134, 6.66062702753093859465403758477, 6.83921001446596737633967757560, 7.41093439925521310147741227555, 8.451991930124608559593159911555, 8.610264838777983408651887538747, 9.103938484655653492425357032401, 9.173706923637329481742301738429, 9.752887239297725819796461544628, 10.36006400201378464173297109921, 10.90494291763676779852377629713, 10.98017512674928301187353752954, 11.92612142471837471706435068419, 12.28832103067981868195069364999
