| L(s) = 1 | − 2·11-s − 4·19-s + 16·31-s + 10·49-s − 24·59-s + 4·61-s + 20·79-s − 12·89-s + 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.917·19-s + 2.87·31-s + 10/7·49-s − 3.12·59-s + 0.512·61-s + 2.25·79-s − 1.27·89-s + 1.91·109-s + 3/11·121-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 + 1.69·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.735165061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.735165061\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−7.907672317322422457940518425226, −7.49805739145605137960613894121, −7.12068215535004271605339690567, −6.87235306385509193419504550801, −6.29569057208847450959258894539, −6.27196307826938811660167650392, −5.84915846113848306575796986464, −5.50765033728964647716048995913, −5.01064359823587162681253443127, −4.61160131762081424225306979587, −4.45333618386164719002048325065, −4.18717709578446036660788496631, −3.42224423859959338781371920448, −3.29722621162095840904822373907, −2.73296211462968906031308802558, −2.45569246631892935879418968511, −2.00165089848070053595455326377, −1.51160609183823835539744806707, −0.826211700614489041243657735083, −0.46233408403600640378579358529,
0.46233408403600640378579358529, 0.826211700614489041243657735083, 1.51160609183823835539744806707, 2.00165089848070053595455326377, 2.45569246631892935879418968511, 2.73296211462968906031308802558, 3.29722621162095840904822373907, 3.42224423859959338781371920448, 4.18717709578446036660788496631, 4.45333618386164719002048325065, 4.61160131762081424225306979587, 5.01064359823587162681253443127, 5.50765033728964647716048995913, 5.84915846113848306575796986464, 6.27196307826938811660167650392, 6.29569057208847450959258894539, 6.87235306385509193419504550801, 7.12068215535004271605339690567, 7.49805739145605137960613894121, 7.907672317322422457940518425226
