| L(s) = 1 | − 120·17-s + 76·25-s + 24·41-s − 4·49-s − 344·73-s − 312·89-s + 248·97-s + 24·113-s − 100·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 580·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 7.05·17-s + 3.03·25-s + 0.585·41-s − 0.0816·49-s − 4.71·73-s − 3.50·89-s + 2.55·97-s + 0.212·113-s − 0.826·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.43·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2935283245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2935283245\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 674 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 914 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1018 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1726 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2638 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5318 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3074 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2}( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 6626 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 6482 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 11038 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13586 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{4} \) |
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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
−7.38893709189746923848017209327, −7.30303014165155537767067964276, −6.86905918201796305504872498717, −6.86218345965042272201075070055, −6.60871020463681693573581388024, −6.35553075534168364422087835387, −6.24655490795604291462452845787, −5.96676204210569846680043446906, −5.58678463337727409717688986445, −5.07651414132702022736546687496, −5.03844751664607348139678014881, −4.53906099579572994552045647724, −4.52484715522237396919404583613, −4.31880862200366076178991506554, −4.24792991917049444724925928247, −3.81276735275334532379363049809, −3.25247017618835364966359363179, −2.86419562919665502986117135091, −2.69489393388727772336867656866, −2.33530356157797945870074939431, −2.27809206313197820495624340122, −1.62186718045250429957183128699, −1.47183051499872596619843961828, −0.61283480987577486908157251457, −0.12566935799982811304978515457,
0.12566935799982811304978515457, 0.61283480987577486908157251457, 1.47183051499872596619843961828, 1.62186718045250429957183128699, 2.27809206313197820495624340122, 2.33530356157797945870074939431, 2.69489393388727772336867656866, 2.86419562919665502986117135091, 3.25247017618835364966359363179, 3.81276735275334532379363049809, 4.24792991917049444724925928247, 4.31880862200366076178991506554, 4.52484715522237396919404583613, 4.53906099579572994552045647724, 5.03844751664607348139678014881, 5.07651414132702022736546687496, 5.58678463337727409717688986445, 5.96676204210569846680043446906, 6.24655490795604291462452845787, 6.35553075534168364422087835387, 6.60871020463681693573581388024, 6.86218345965042272201075070055, 6.86905918201796305504872498717, 7.30303014165155537767067964276, 7.38893709189746923848017209327
