| L(s) = 1 | − 44·13-s − 50·25-s − 52·37-s − 94·49-s − 148·61-s + 92·73-s − 4·97-s − 428·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.11e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.38·13-s − 2·25-s − 1.40·37-s − 1.91·49-s − 2.42·61-s + 1.26·73-s − 0.0412·97-s − 3.92·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.59·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1898039236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1898039236\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 122 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
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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
−11.18972923320816084208860155527, −10.30357413746711068333346277250, −9.706253806604030231992406785358, −9.432943737615824419949993019392, −9.381165707856235023781619948394, −8.387764700831519311143484577977, −8.028242941156895393206216269965, −7.45608064376289718787848283260, −7.40701078219997835513883553352, −6.69649500606453731779975566609, −6.31140055277020989766433244611, −5.40762948499913935358358810103, −5.30081113960464534470279964269, −4.62243142644478892632395509177, −4.29058232889079044615044716977, −3.41729449384751515787207582216, −2.84922021075563488707169396731, −2.17983975014776508994410069258, −1.71106739352607791725476065679, −0.15444809734092919225190994511,
0.15444809734092919225190994511, 1.71106739352607791725476065679, 2.17983975014776508994410069258, 2.84922021075563488707169396731, 3.41729449384751515787207582216, 4.29058232889079044615044716977, 4.62243142644478892632395509177, 5.30081113960464534470279964269, 5.40762948499913935358358810103, 6.31140055277020989766433244611, 6.69649500606453731779975566609, 7.40701078219997835513883553352, 7.45608064376289718787848283260, 8.028242941156895393206216269965, 8.387764700831519311143484577977, 9.381165707856235023781619948394, 9.432943737615824419949993019392, 9.706253806604030231992406785358, 10.30357413746711068333346277250, 11.18972923320816084208860155527
