| L(s) = 1 | − 2·5-s + 9-s + 3·25-s − 2·29-s + 2·41-s − 2·45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s + 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·5-s + 9-s + 3·25-s − 2·29-s + 2·41-s − 2·45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s + 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9032356442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9032356442\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.670518571064718181213019787662, −8.468516270749958112318313506889, −7.948908863248564490121492483308, −7.76655971803882216531330490111, −7.35407386036474982897199637872, −7.06217667125121731686400338468, −6.92373550296238589167841172083, −6.38169813902501019082743582940, −5.68223021165370224851472751190, −5.59242626274693491723804348987, −4.95782050175129613019179083427, −4.47862804396335827529305299116, −4.21143610443135802774432277955, −3.94997644358720342591134005476, −3.52953135609114500770691446304, −3.10620335820860592271843950989, −2.54316683244857705306886928076, −1.95033197646336209607277874156, −1.25704503398957430265168342870, −0.59689074176784338880675773165,
0.59689074176784338880675773165, 1.25704503398957430265168342870, 1.95033197646336209607277874156, 2.54316683244857705306886928076, 3.10620335820860592271843950989, 3.52953135609114500770691446304, 3.94997644358720342591134005476, 4.21143610443135802774432277955, 4.47862804396335827529305299116, 4.95782050175129613019179083427, 5.59242626274693491723804348987, 5.68223021165370224851472751190, 6.38169813902501019082743582940, 6.92373550296238589167841172083, 7.06217667125121731686400338468, 7.35407386036474982897199637872, 7.76655971803882216531330490111, 7.948908863248564490121492483308, 8.468516270749958112318313506889, 8.670518571064718181213019787662
