| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 4·10-s + 6·11-s − 6·12-s − 4·14-s + 4·15-s + 5·16-s − 6·17-s + 6·18-s − 10·19-s − 6·20-s + 4·21-s + 12·22-s + 2·23-s − 8·24-s + 3·25-s − 4·27-s − 6·28-s − 6·29-s + 8·30-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 + 9-s − 1.26·10-s + 1.80·11-s − 1.73·12-s − 1.06·14-s + 1.03·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s − 2.29·19-s − 1.34·20-s + 0.872·21-s + 2.55·22-s + 0.417·23-s − 1.63·24-s + 3/5·25-s − 0.769·27-s − 1.13·28-s − 1.11·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−7.70063714530740740498931368302, −7.64399277146109829388062093444, −7.07332781314437348059983579182, −6.88425055285990137837383893323, −6.38739931523244738129578877265, −6.35239941375921154994827483246, −5.93442510795810601035345900917, −5.76164489745189575599858317277, −4.89634122864822599881260298884, −4.71807725384004305443602300758, −4.23158041964397187708297801504, −4.22648486427376821804230451787, −3.64759967828559953383731180818, −3.50864170549467983520500156700, −2.56557523822428985220504576151, −2.46312219072356168256271909699, −1.45273844681560206678232182550, −1.40835197943441898555913567928, 0, 0,
1.40835197943441898555913567928, 1.45273844681560206678232182550, 2.46312219072356168256271909699, 2.56557523822428985220504576151, 3.50864170549467983520500156700, 3.64759967828559953383731180818, 4.22648486427376821804230451787, 4.23158041964397187708297801504, 4.71807725384004305443602300758, 4.89634122864822599881260298884, 5.76164489745189575599858317277, 5.93442510795810601035345900917, 6.35239941375921154994827483246, 6.38739931523244738129578877265, 6.88425055285990137837383893323, 7.07332781314437348059983579182, 7.64399277146109829388062093444, 7.70063714530740740498931368302
