| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·13-s + 2·15-s + 4·17-s − 8·19-s − 25-s − 27-s + 12·29-s + 12·37-s + 4·39-s − 2·45-s − 14·49-s − 4·51-s + 8·57-s + 8·65-s + 16·71-s + 75-s + 81-s − 8·83-s − 8·85-s − 12·87-s + 16·95-s − 36·101-s + 32·103-s − 24·107-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.10·13-s + 0.516·15-s + 0.970·17-s − 1.83·19-s − 1/5·25-s − 0.192·27-s + 2.22·29-s + 1.97·37-s + 0.640·39-s − 0.298·45-s − 2·49-s − 0.560·51-s + 1.05·57-s + 0.992·65-s + 1.89·71-s + 0.115·75-s + 1/9·81-s − 0.878·83-s − 0.867·85-s − 1.28·87-s + 1.64·95-s − 3.58·101-s + 3.15·103-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.255796980767850970322001251893, −8.098990694093691505068092710868, −7.67736253338048188697283010037, −7.11232565976809397151828301173, −6.42897107072744719896577758454, −6.42343544296194884917141468428, −5.64987942678297537611315366523, −5.01003276702092665382516487483, −4.54865315907754866949539090997, −4.25303028692796488061253338187, −3.53710037747194767985133374261, −2.80568477694902190771689458066, −2.21845693694876550409461648737, −1.08177483520843276045938558632, 0,
1.08177483520843276045938558632, 2.21845693694876550409461648737, 2.80568477694902190771689458066, 3.53710037747194767985133374261, 4.25303028692796488061253338187, 4.54865315907754866949539090997, 5.01003276702092665382516487483, 5.64987942678297537611315366523, 6.42343544296194884917141468428, 6.42897107072744719896577758454, 7.11232565976809397151828301173, 7.67736253338048188697283010037, 8.098990694093691505068092710868, 8.255796980767850970322001251893
