| L(s) = 1 | − 8·7-s + 4·13-s + 16·19-s − 10·25-s − 8·31-s − 20·37-s + 16·43-s + 34·49-s + 28·61-s − 32·67-s − 20·73-s − 8·79-s − 32·91-s + 28·97-s + 40·103-s + 4·109-s − 22·121-s + 127-s + 131-s − 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 1.10·13-s + 3.67·19-s − 2·25-s − 1.43·31-s − 3.28·37-s + 2.43·43-s + 34/7·49-s + 3.58·61-s − 3.90·67-s − 2.34·73-s − 0.900·79-s − 3.35·91-s + 2.84·97-s + 3.94·103-s + 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 11.0·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4915286641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4915286641\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.74858543777126986291158021371, −18.85850998760573807432087794673, −18.85850998760573807432087794673, −17.68732582410033881966766579011, −17.68732582410033881966766579011, −16.25038600345248605266882134733, −16.25038600345248605266882134733, −15.69696813163500519905091038725, −15.69696813163500519905091038725, −13.99634105119279079569568395122, −13.99634105119279079569568395122, −13.01055982622439917603548235428, −13.01055982622439917603548235428, −11.77437667375267836950691610862, −11.77437667375267836950691610862, −10.17441103098667470227930585744, −10.17441103098667470227930585744, −9.113424945499136957715665264292, −9.113424945499136957715665264292, −7.26646731082131852272265775116, −7.26646731082131852272265775116, −5.80268955254619590131632024787, −5.80268955254619590131632024787, −3.44334336790947687892993758586, −3.44334336790947687892993758586,
3.44334336790947687892993758586, 3.44334336790947687892993758586, 5.80268955254619590131632024787, 5.80268955254619590131632024787, 7.26646731082131852272265775116, 7.26646731082131852272265775116, 9.113424945499136957715665264292, 9.113424945499136957715665264292, 10.17441103098667470227930585744, 10.17441103098667470227930585744, 11.77437667375267836950691610862, 11.77437667375267836950691610862, 13.01055982622439917603548235428, 13.01055982622439917603548235428, 13.99634105119279079569568395122, 13.99634105119279079569568395122, 15.69696813163500519905091038725, 15.69696813163500519905091038725, 16.25038600345248605266882134733, 16.25038600345248605266882134733, 17.68732582410033881966766579011, 17.68732582410033881966766579011, 18.85850998760573807432087794673, 18.85850998760573807432087794673, 19.74858543777126986291158021371
