| L(s) = 1 | − 4·3-s − 2·9-s + 12·23-s − 4·25-s + 40·27-s − 12·29-s − 12·41-s + 36·47-s + 28·49-s − 48·69-s + 16·75-s − 55·81-s + 48·87-s + 48·101-s + 4·121-s + 48·123-s + 127-s + 131-s + 137-s + 139-s − 144·141-s − 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2/3·9-s + 2.50·23-s − 4/5·25-s + 7.69·27-s − 2.22·29-s − 1.87·41-s + 5.25·47-s + 4·49-s − 5.77·69-s + 1.84·75-s − 6.11·81-s + 5.14·87-s + 4.77·101-s + 4/11·121-s + 4.32·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 12.1·141-s − 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3985007068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3985007068\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.54595771715411699149475295052, −6.20739285803104364496575028317, −5.90857152731437429865548924810, −5.84045281471823175332670279333, −5.76134302265987595293820498578, −5.55426292876499768277845738675, −5.51067007451890488921641209216, −5.16056666922376986778161543478, −5.13834659140349665586228058570, −4.72681736403762576422305016699, −4.52120335406798510341075795911, −4.25885469736385794446059668800, −4.12337135156646103650503542461, −3.51864831528984031654105122518, −3.47630767767347573219527445396, −3.23218360042330779864872587638, −3.06603672275557544851604930485, −2.58088685929816378958897012716, −2.33621672628916130765958438838, −2.20348192782250517393963725080, −2.00174217957129748390936091453, −1.01524059167307022167846659147, −0.875705896901860638765566280299, −0.803299423139157110648410327211, −0.20962875270786755658855976663,
0.20962875270786755658855976663, 0.803299423139157110648410327211, 0.875705896901860638765566280299, 1.01524059167307022167846659147, 2.00174217957129748390936091453, 2.20348192782250517393963725080, 2.33621672628916130765958438838, 2.58088685929816378958897012716, 3.06603672275557544851604930485, 3.23218360042330779864872587638, 3.47630767767347573219527445396, 3.51864831528984031654105122518, 4.12337135156646103650503542461, 4.25885469736385794446059668800, 4.52120335406798510341075795911, 4.72681736403762576422305016699, 5.13834659140349665586228058570, 5.16056666922376986778161543478, 5.51067007451890488921641209216, 5.55426292876499768277845738675, 5.76134302265987595293820498578, 5.84045281471823175332670279333, 5.90857152731437429865548924810, 6.20739285803104364496575028317, 6.54595771715411699149475295052
