| L(s) = 1 | − 2·4-s − 4·7-s + 4·11-s + 3·16-s − 2·25-s + 8·28-s + 2·37-s − 14·41-s − 8·44-s − 18·49-s + 8·53-s − 4·64-s + 4·67-s + 24·71-s + 18·73-s − 16·77-s − 10·83-s + 4·100-s + 12·101-s + 14·107-s − 12·112-s − 34·121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 1.20·11-s + 3/4·16-s − 2/5·25-s + 1.51·28-s + 0.328·37-s − 2.18·41-s − 1.20·44-s − 2.57·49-s + 1.09·53-s − 1/2·64-s + 0.488·67-s + 2.84·71-s + 2.10·73-s − 1.82·77-s − 1.09·83-s + 2/5·100-s + 1.19·101-s + 1.35·107-s − 1.13·112-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.328·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4045446473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4045446473\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 15 T^{2} + 376 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 15 T^{2} - 116 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 404 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 55 T^{2} + 1544 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 63 T^{2} + 2020 T^{4} - 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 7 T + 76 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 75 T^{2} + 2896 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 8606 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 63 T^{2} + 3160 T^{4} - 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 9 T + 148 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 152 T^{2} + 15630 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 5 T + 154 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 13 T^{2} + 2580 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 267 T^{2} + 33556 T^{4} - 267 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.18421010551790684908672606454, −5.88227541493848101245801609164, −5.70944635047447525128481600264, −5.58780656960811129490313654300, −5.27180845095302108205600725340, −4.94033382549607721366142257581, −4.93598913306011564123489085070, −4.78157744838321647734294124679, −4.47895403829506723129423182779, −4.23046028899371530564944203745, −4.00571812639642856180300766667, −3.68038106460513237966599122867, −3.66848253610843747969761339479, −3.41676633444817839873570090198, −3.32561831312769064256456780318, −2.96076826390801111928409686114, −2.90901306150779596697702020260, −2.34695967062498802606730096442, −2.07893625194029006023946481138, −2.03141151084177027312994500709, −1.50652441954701330223800895279, −1.39274435784194112051589470155, −0.75573437583752738728305891077, −0.73784531045714428407572648334, −0.12543464394368656207169685668,
0.12543464394368656207169685668, 0.73784531045714428407572648334, 0.75573437583752738728305891077, 1.39274435784194112051589470155, 1.50652441954701330223800895279, 2.03141151084177027312994500709, 2.07893625194029006023946481138, 2.34695967062498802606730096442, 2.90901306150779596697702020260, 2.96076826390801111928409686114, 3.32561831312769064256456780318, 3.41676633444817839873570090198, 3.66848253610843747969761339479, 3.68038106460513237966599122867, 4.00571812639642856180300766667, 4.23046028899371530564944203745, 4.47895403829506723129423182779, 4.78157744838321647734294124679, 4.93598913306011564123489085070, 4.94033382549607721366142257581, 5.27180845095302108205600725340, 5.58780656960811129490313654300, 5.70944635047447525128481600264, 5.88227541493848101245801609164, 6.18421010551790684908672606454
