| L(s) = 1 | − 4·4-s + 4·7-s + 12·16-s + 2·25-s − 16·28-s − 40·43-s + 10·49-s + 56·61-s − 32·64-s + 8·67-s − 8·100-s − 16·103-s + 56·109-s + 48·112-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 160·172-s + 173-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.51·7-s + 3·16-s + 2/5·25-s − 3.02·28-s − 6.09·43-s + 10/7·49-s + 7.17·61-s − 4·64-s + 0.977·67-s − 4/5·100-s − 1.57·103-s + 5.36·109-s + 4.53·112-s − 1.81·121-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 + 3.07·169-s + 12.1·172-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011621783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011621783\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 188 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.86723758353901322590105352942, −6.70076489433305701530355943702, −6.66571270765553567977344231825, −6.22271908875392212150937733663, −5.87497507414021054717504556016, −5.77114300180328126491493088507, −5.39781850418422716357582848268, −5.12635795993032034886107948227, −5.05034481659475778249410216063, −4.96454044222577409254578726463, −4.95294760888753688727276227100, −4.48243892711722308777249992453, −4.01999276328969179993417449405, −3.99578511385983102650997205984, −3.87015197655782717023048209341, −3.50695947202046510528366667124, −3.15352675575213029136399173154, −3.09948145954682040498986623490, −2.62786515649951401805281758735, −1.92468943676297206586902109869, −1.90869673011106777398476549036, −1.82982542507685073972916277570, −1.02113967179276266090583070717, −0.869239023628722976127759342948, −0.39672638214439083478024327802,
0.39672638214439083478024327802, 0.869239023628722976127759342948, 1.02113967179276266090583070717, 1.82982542507685073972916277570, 1.90869673011106777398476549036, 1.92468943676297206586902109869, 2.62786515649951401805281758735, 3.09948145954682040498986623490, 3.15352675575213029136399173154, 3.50695947202046510528366667124, 3.87015197655782717023048209341, 3.99578511385983102650997205984, 4.01999276328969179993417449405, 4.48243892711722308777249992453, 4.95294760888753688727276227100, 4.96454044222577409254578726463, 5.05034481659475778249410216063, 5.12635795993032034886107948227, 5.39781850418422716357582848268, 5.77114300180328126491493088507, 5.87497507414021054717504556016, 6.22271908875392212150937733663, 6.66571270765553567977344231825, 6.70076489433305701530355943702, 6.86723758353901322590105352942
