| L(s) = 1 | + 4·13-s − 2·25-s − 28·37-s + 22·49-s − 4·61-s + 4·73-s + 4·97-s + 40·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 2/5·25-s − 4.60·37-s + 22/7·49-s − 0.512·61-s + 0.468·73-s + 0.406·97-s + 3.83·109-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 − 3.23·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5706442043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5706442043\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 107 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 83 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
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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
−6.33099047397905100550774477181, −6.23624482994640978458263359942, −5.99532774583619869436576379465, −5.94745949388314923772472107160, −5.52906071721795402969418661162, −5.47426832600293440783023661910, −5.10411608875379806072954311305, −5.07804056398547159204739192426, −4.81598241870371231887458899593, −4.58399955471075539385164287439, −4.24107494207845485717800315905, −3.94408378957739154941273383026, −3.85203247271678446037838200895, −3.54559552449163762711873395931, −3.49633787112694568590308516523, −3.27462571662969934193773560785, −2.77045584621031606995033385689, −2.73080153803482116321796042247, −2.14178722861798386979094389402, −2.13694360238549647192676374937, −1.75796999180877063508061669009, −1.54165185789996133310112757407, −0.963429626694168764341601162701, −0.937952479543584697810901185521, −0.12976108285368816478317225020,
0.12976108285368816478317225020, 0.937952479543584697810901185521, 0.963429626694168764341601162701, 1.54165185789996133310112757407, 1.75796999180877063508061669009, 2.13694360238549647192676374937, 2.14178722861798386979094389402, 2.73080153803482116321796042247, 2.77045584621031606995033385689, 3.27462571662969934193773560785, 3.49633787112694568590308516523, 3.54559552449163762711873395931, 3.85203247271678446037838200895, 3.94408378957739154941273383026, 4.24107494207845485717800315905, 4.58399955471075539385164287439, 4.81598241870371231887458899593, 5.07804056398547159204739192426, 5.10411608875379806072954311305, 5.47426832600293440783023661910, 5.52906071721795402969418661162, 5.94745949388314923772472107160, 5.99532774583619869436576379465, 6.23624482994640978458263359942, 6.33099047397905100550774477181
