| L(s) = 1 | + 12·9-s − 20·25-s − 14·49-s + 90·81-s + 8·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 240·225-s + 227-s + ⋯ |
| L(s) = 1 | + 4·9-s − 4·25-s − 2·49-s + 10·81-s + 0.752·113-s + 1.09·121-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 − 4·169-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 + 0.0669·223-s − 16·225-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.438017682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.438017682\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.71497343406342298249967705676, −6.33246660329563653551846667278, −6.27661407144869441251839686396, −6.15730352341338052470420320490, −5.74823144586505082747215942590, −5.44541766318167866861287528740, −5.41043382503614681747357274451, −4.87810160201851801591789898723, −4.85947198389806214212874203375, −4.70751578525888815984967594473, −4.26681220512043031696503215102, −4.13324079529668552505420176572, −4.12250871164303618833151258929, −3.65191511571204231747643109576, −3.64346788639910804449702837756, −3.51538272761900889906598995391, −2.96267474376401946616515117236, −2.62643578659415636288269685106, −2.26272384328818931489374653204, −1.89755061656783189113881381262, −1.79450059897825166281683950311, −1.64269119880297867522777089324, −1.29806902455609965355266323446, −0.891045092888046737593636769054, −0.32399554543082494128334546995,
0.32399554543082494128334546995, 0.891045092888046737593636769054, 1.29806902455609965355266323446, 1.64269119880297867522777089324, 1.79450059897825166281683950311, 1.89755061656783189113881381262, 2.26272384328818931489374653204, 2.62643578659415636288269685106, 2.96267474376401946616515117236, 3.51538272761900889906598995391, 3.64346788639910804449702837756, 3.65191511571204231747643109576, 4.12250871164303618833151258929, 4.13324079529668552505420176572, 4.26681220512043031696503215102, 4.70751578525888815984967594473, 4.85947198389806214212874203375, 4.87810160201851801591789898723, 5.41043382503614681747357274451, 5.44541766318167866861287528740, 5.74823144586505082747215942590, 6.15730352341338052470420320490, 6.27661407144869441251839686396, 6.33246660329563653551846667278, 6.71497343406342298249967705676
