| L(s) = 1 | + 6·11-s + 16·19-s − 6·29-s − 10·31-s + 6·41-s − 13·49-s + 6·59-s + 26·61-s + 48·71-s + 22·79-s + 24·89-s + 30·101-s − 8·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.80·11-s + 3.67·19-s − 1.11·29-s − 1.79·31-s + 0.937·41-s − 1.85·49-s + 0.781·59-s + 3.32·61-s + 5.69·71-s + 2.47·79-s + 2.54·89-s + 2.98·101-s − 0.766·109-s + 2.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.597639101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.597639101\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 73 T^{2} - 4080 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.41472476230803739521172811380, −5.98416561706554884011410879445, −5.86411383093590383064641348072, −5.60246734878673315859947787575, −5.48720345880660625204335646196, −5.08934890088221056649849747497, −5.02705321673935425612384728759, −4.95650211689555763466934952526, −4.91466259543318138811294963548, −4.18171369751995756842290522315, −4.08586333565605284053376950274, −3.86060785697906169053930440476, −3.75643694598376137151292525559, −3.47727373942673935744038169595, −3.28155668170697041361954270769, −3.12368535933943603941275638930, −3.06484990658246662407741500861, −2.23674300011553588499729938641, −2.14407016360217130256172821750, −2.03067928753312742002325838902, −1.86912232728428613208869045672, −1.26643466597662421878467426598, −0.883927912402901540319852003873, −0.798899682111628645875452865526, −0.61676624764186247690829387859,
0.61676624764186247690829387859, 0.798899682111628645875452865526, 0.883927912402901540319852003873, 1.26643466597662421878467426598, 1.86912232728428613208869045672, 2.03067928753312742002325838902, 2.14407016360217130256172821750, 2.23674300011553588499729938641, 3.06484990658246662407741500861, 3.12368535933943603941275638930, 3.28155668170697041361954270769, 3.47727373942673935744038169595, 3.75643694598376137151292525559, 3.86060785697906169053930440476, 4.08586333565605284053376950274, 4.18171369751995756842290522315, 4.91466259543318138811294963548, 4.95650211689555763466934952526, 5.02705321673935425612384728759, 5.08934890088221056649849747497, 5.48720345880660625204335646196, 5.60246734878673315859947787575, 5.86411383093590383064641348072, 5.98416561706554884011410879445, 6.41472476230803739521172811380
