| L(s) = 1 | − 2-s + 2·4-s − 2·7-s − 5·8-s + 9-s + 4·11-s + 2·14-s + 5·16-s − 18-s − 4·22-s + 10·25-s − 4·28-s − 10·32-s + 2·36-s + 16·37-s + 8·43-s + 8·44-s − 11·49-s − 10·50-s + 8·53-s + 10·56-s − 2·63-s + 17·64-s − 5·72-s − 16·74-s − 8·77-s + 26·79-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 0.755·7-s − 1.76·8-s + 1/3·9-s + 1.20·11-s + 0.534·14-s + 5/4·16-s − 0.235·18-s − 0.852·22-s + 2·25-s − 0.755·28-s − 1.76·32-s + 1/3·36-s + 2.63·37-s + 1.21·43-s + 1.20·44-s − 1.57·49-s − 1.41·50-s + 1.09·53-s + 1.33·56-s − 0.251·63-s + 17/8·64-s − 0.589·72-s − 1.85·74-s − 0.911·77-s + 2.92·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631717314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631717314\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 6 T^{2} - 253 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 50 T^{2} + 1539 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 111 T^{2} + 8840 T^{4} + 111 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 115 T^{2} + 9504 T^{4} + 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 + 127 T^{2} + 11640 T^{4} + 127 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 34 T^{2} - 4173 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.688106580529564596149193465987, −8.254000898637319667647642313932, −7.939997294723289162588479863487, −7.69520832298764954453756809465, −7.69292817981257631736683171156, −7.04694028035111952244533651038, −7.02205138122538449900217255525, −6.55452601601042274363952623370, −6.54056845175998895519120571389, −6.24119971776377700456157998315, −6.17491161113186027297624416531, −5.77926771877885086278316910738, −5.28096265557092402268158243804, −5.15927870358845290077239650599, −4.73289205300307711125971517374, −4.30139628312825422825387016938, −4.10499508352843310201253655710, −3.63015172122245793587229739462, −3.23869005904284853338157907331, −3.16676602594659353979993680610, −2.68344781015173082511399528384, −2.16637211343685779202692696727, −2.05172014348148192776193157999, −0.985205700718565601513783445839, −0.855701828923577491365219203156,
0.855701828923577491365219203156, 0.985205700718565601513783445839, 2.05172014348148192776193157999, 2.16637211343685779202692696727, 2.68344781015173082511399528384, 3.16676602594659353979993680610, 3.23869005904284853338157907331, 3.63015172122245793587229739462, 4.10499508352843310201253655710, 4.30139628312825422825387016938, 4.73289205300307711125971517374, 5.15927870358845290077239650599, 5.28096265557092402268158243804, 5.77926771877885086278316910738, 6.17491161113186027297624416531, 6.24119971776377700456157998315, 6.54056845175998895519120571389, 6.55452601601042274363952623370, 7.02205138122538449900217255525, 7.04694028035111952244533651038, 7.69292817981257631736683171156, 7.69520832298764954453756809465, 7.939997294723289162588479863487, 8.254000898637319667647642313932, 8.688106580529564596149193465987
