L(s) = 1 | + 3·2-s + 2·3-s + 4·4-s + 6·6-s + 2·7-s + 3·8-s − 3·9-s − 2·11-s + 8·12-s + 8·13-s + 6·14-s + 3·16-s + 4·17-s − 9·18-s + 2·19-s + 4·21-s − 6·22-s + 2·23-s + 6·24-s + 24·26-s − 14·27-s + 8·28-s − 10·29-s + 4·31-s + 6·32-s − 4·33-s + 12·34-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 1.15·3-s + 2·4-s + 2.44·6-s + 0.755·7-s + 1.06·8-s − 9-s − 0.603·11-s + 2.30·12-s + 2.21·13-s + 1.60·14-s + 3/4·16-s + 0.970·17-s − 2.12·18-s + 0.458·19-s + 0.872·21-s − 1.27·22-s + 0.417·23-s + 1.22·24-s + 4.70·26-s − 2.69·27-s + 1.51·28-s − 1.85·29-s + 0.718·31-s + 1.06·32-s − 0.696·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.542132585\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.542132585\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 274 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−11.10911794872411536474810803491, −10.79665287608434845988667564892, −10.21328943056263325275742851703, −9.507468800920776256282332704371, −9.063656034039517945008518621852, −8.703618793056692228215169577434, −8.014013849197658166476626857420, −7.997255849139558219268896890065, −7.58595103679307296476772484357, −6.59145076816759939765121746332, −6.11036377884088353359100696984, −5.69954350468233113796506375934, −5.27294936805960046467214722930, −5.05762884202645613650587765067, −3.95679090222986520805569708975, −3.94719273499862893396810960549, −3.13516082241924280439628057420, −3.11800972478402242933589508765, −2.19820201054592301214911492646, −1.31465765825216604923597806682,
1.31465765825216604923597806682, 2.19820201054592301214911492646, 3.11800972478402242933589508765, 3.13516082241924280439628057420, 3.94719273499862893396810960549, 3.95679090222986520805569708975, 5.05762884202645613650587765067, 5.27294936805960046467214722930, 5.69954350468233113796506375934, 6.11036377884088353359100696984, 6.59145076816759939765121746332, 7.58595103679307296476772484357, 7.997255849139558219268896890065, 8.014013849197658166476626857420, 8.703618793056692228215169577434, 9.063656034039517945008518621852, 9.507468800920776256282332704371, 10.21328943056263325275742851703, 10.79665287608434845988667564892, 11.10911794872411536474810803491