| L(s) = 1 | + 2·13-s + 4·37-s − 49-s − 2·61-s − 4·73-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2·13-s + 4·37-s − 49-s − 2·61-s − 4·73-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.701829136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.701829136\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−8.867470330904173342821232056490, −8.446156141950173316095183383653, −8.352937788875759211196280498756, −7.74150735439921101569748939819, −7.39502693616187491454632631477, −7.34431156178424980269602863901, −6.41932832409070900917269557885, −6.25529837166572875014060175006, −5.96532382474648780660299390712, −5.85717804278547754481236016725, −5.10156698455657904980448646174, −4.53293702398748791021384047300, −4.41151213172758678057851222304, −3.96765265705382171201653111116, −3.35147616920509415702730398561, −3.06362229102243986605066833457, −2.61659039085290545781143792099, −1.88976216955315927353710890275, −1.36858935655896603797884666969, −0.849287628200568591356291987110,
0.849287628200568591356291987110, 1.36858935655896603797884666969, 1.88976216955315927353710890275, 2.61659039085290545781143792099, 3.06362229102243986605066833457, 3.35147616920509415702730398561, 3.96765265705382171201653111116, 4.41151213172758678057851222304, 4.53293702398748791021384047300, 5.10156698455657904980448646174, 5.85717804278547754481236016725, 5.96532382474648780660299390712, 6.25529837166572875014060175006, 6.41932832409070900917269557885, 7.34431156178424980269602863901, 7.39502693616187491454632631477, 7.74150735439921101569748939819, 8.352937788875759211196280498756, 8.446156141950173316095183383653, 8.867470330904173342821232056490
