| L(s) = 1 | − 3·9-s + 12·11-s − 28·49-s − 48·59-s + 32·61-s − 24·71-s − 36·99-s + 32·109-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 9-s + 3.61·11-s − 4·49-s − 6.24·59-s + 4.09·61-s − 2.84·71-s − 3.61·99-s + 3.06·109-s + 4.18·121-s + 0.0887·127-s + 0.0873·131-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 + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.001264976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.001264976\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.82831561822013082708047393265, −6.70342598585781322552192874014, −6.38493409967277455843294220206, −6.29091755807794665206228127158, −6.26480525978599947341167265002, −6.08549574358512315846666321331, −5.57897812697135616083749446623, −5.51914440579587550867725907657, −5.21136402672063117024022424190, −4.75953269564497565801448055856, −4.75403945014907289465644059732, −4.35118357212342791921834186669, −4.16793754532892976921673650028, −4.10175045680120140708832595382, −3.73295705856304669040381883880, −3.25345264355265023227951918481, −3.18393296070522208788095908421, −3.01048852668052080950416615439, −2.92375189151499788196178862860, −2.07885672709041238434560069773, −1.82032635339650145079287179848, −1.56978298991214834476350421410, −1.53517393047096860460140843450, −0.870554003498343758620055650610, −0.40257332936735808558594623100,
0.40257332936735808558594623100, 0.870554003498343758620055650610, 1.53517393047096860460140843450, 1.56978298991214834476350421410, 1.82032635339650145079287179848, 2.07885672709041238434560069773, 2.92375189151499788196178862860, 3.01048852668052080950416615439, 3.18393296070522208788095908421, 3.25345264355265023227951918481, 3.73295705856304669040381883880, 4.10175045680120140708832595382, 4.16793754532892976921673650028, 4.35118357212342791921834186669, 4.75403945014907289465644059732, 4.75953269564497565801448055856, 5.21136402672063117024022424190, 5.51914440579587550867725907657, 5.57897812697135616083749446623, 6.08549574358512315846666321331, 6.26480525978599947341167265002, 6.29091755807794665206228127158, 6.38493409967277455843294220206, 6.70342598585781322552192874014, 6.82831561822013082708047393265
