| L(s) = 1 | + 2·3-s − 2·5-s − 7-s − 2·9-s + 2·13-s − 4·15-s + 8·17-s − 2·19-s − 2·21-s − 6·25-s − 10·27-s − 16·29-s + 4·31-s + 2·35-s + 4·39-s + 16·41-s − 8·43-s + 4·45-s + 12·47-s − 6·49-s + 16·51-s + 20·53-s − 4·57-s + 6·59-s − 2·61-s + 2·63-s − 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s + 0.554·13-s − 1.03·15-s + 1.94·17-s − 0.458·19-s − 0.436·21-s − 6/5·25-s − 1.92·27-s − 2.97·29-s + 0.718·31-s + 0.338·35-s + 0.640·39-s + 2.49·41-s − 1.21·43-s + 0.596·45-s + 1.75·47-s − 6/7·49-s + 2.24·51-s + 2.74·53-s − 0.529·57-s + 0.781·59-s − 0.256·61-s + 0.251·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8688785795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8688785795\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−18.1183629461, −17.3712810614, −16.7385636699, −16.5496846221, −15.7488207479, −15.2874088787, −14.6041850008, −14.5765256398, −13.7811815886, −13.2768755254, −12.7433993225, −11.7666126827, −11.7106476104, −10.9076921437, −10.1146500656, −9.39249415773, −8.95538623117, −8.18264311684, −7.77199473906, −7.29668061465, −5.87146418849, −5.62168100150, −3.82418745638, −3.67478222653, −2.48514181669,
2.48514181669, 3.67478222653, 3.82418745638, 5.62168100150, 5.87146418849, 7.29668061465, 7.77199473906, 8.18264311684, 8.95538623117, 9.39249415773, 10.1146500656, 10.9076921437, 11.7106476104, 11.7666126827, 12.7433993225, 13.2768755254, 13.7811815886, 14.5765256398, 14.6041850008, 15.2874088787, 15.7488207479, 16.5496846221, 16.7385636699, 17.3712810614, 18.1183629461
