| L(s) = 1 | + 2·5-s − 9-s + 25-s − 4·29-s + 4·41-s − 2·45-s + 49-s − 2·61-s + 81-s + 2·89-s + 2·101-s − 2·109-s − 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2·5-s − 9-s + 25-s − 4·29-s + 4·41-s − 2·45-s + 49-s − 2·61-s + 81-s + 2·89-s + 2·101-s − 2·109-s − 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7773539457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7773539457\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−7.86614928068396116444057771731, −7.73817824788006451697419326899, −7.73259036594636184488012975004, −7.26702222242747083871665126592, −7.18252502849567000973489341711, −6.62316452598821405034664398445, −6.40252308270542119281461890621, −6.33210415928402540225887054118, −5.92209932762572805333585589650, −5.69922555278521477478513933344, −5.68149038148112441402137672218, −5.49400628669275018076183489225, −5.30827537287805132407581586852, −4.76452794896121707658351933171, −4.52521211089171466808079360822, −4.27225319865877781538750414638, −3.82425520553671718166084545484, −3.57956012057656180467308860837, −3.44145577226018232264669468676, −2.84049898887639670279557810562, −2.40141446604507250456079340840, −2.39048707730289601442247771640, −2.07368553047351004769500475548, −1.60945424140335401371850451392, −1.15755955852674574951901317778,
1.15755955852674574951901317778, 1.60945424140335401371850451392, 2.07368553047351004769500475548, 2.39048707730289601442247771640, 2.40141446604507250456079340840, 2.84049898887639670279557810562, 3.44145577226018232264669468676, 3.57956012057656180467308860837, 3.82425520553671718166084545484, 4.27225319865877781538750414638, 4.52521211089171466808079360822, 4.76452794896121707658351933171, 5.30827537287805132407581586852, 5.49400628669275018076183489225, 5.68149038148112441402137672218, 5.69922555278521477478513933344, 5.92209932762572805333585589650, 6.33210415928402540225887054118, 6.40252308270542119281461890621, 6.62316452598821405034664398445, 7.18252502849567000973489341711, 7.26702222242747083871665126592, 7.73259036594636184488012975004, 7.73817824788006451697419326899, 7.86614928068396116444057771731
