| L(s) = 1 | + 2-s + 5-s − 7-s − 8-s + 10-s − 14-s − 16-s − 23-s + 2·29-s − 35-s − 40-s + 2·41-s − 2·43-s − 46-s + 2·47-s + 56-s + 2·58-s + 61-s + 64-s + 67-s − 70-s − 80-s + 2·82-s + 2·83-s − 2·86-s − 89-s + 2·94-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s − 7-s − 8-s + 10-s − 14-s − 16-s − 23-s + 2·29-s − 35-s − 40-s + 2·41-s − 2·43-s − 46-s + 2·47-s + 56-s + 2·58-s + 61-s + 64-s + 67-s − 70-s − 80-s + 2·82-s + 2·83-s − 2·86-s − 89-s + 2·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.541733332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541733332\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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
−10.03041287825331723519172037823, −9.822930528507766646248486705278, −9.301155398737272617678115440260, −9.006437875675312357766268770254, −8.508220377661425978931094521736, −8.137322336826172499067434857226, −7.62570657195080886196752610122, −6.90475128911435753430256383358, −6.65319570068889537034723240009, −6.18212038632062151067222641987, −5.98739279444940651558263870138, −5.42255464932110506178581879258, −5.16209333730369272874811455028, −4.32497573840286987182045922892, −4.27971688913217745042290005954, −3.46999721509126782481426600494, −3.15448821753877464630415015484, −2.41959612521926107032083498081, −2.17305371318448000894708330527, −0.948213553351345036608728533347,
0.948213553351345036608728533347, 2.17305371318448000894708330527, 2.41959612521926107032083498081, 3.15448821753877464630415015484, 3.46999721509126782481426600494, 4.27971688913217745042290005954, 4.32497573840286987182045922892, 5.16209333730369272874811455028, 5.42255464932110506178581879258, 5.98739279444940651558263870138, 6.18212038632062151067222641987, 6.65319570068889537034723240009, 6.90475128911435753430256383358, 7.62570657195080886196752610122, 8.137322336826172499067434857226, 8.508220377661425978931094521736, 9.006437875675312357766268770254, 9.301155398737272617678115440260, 9.822930528507766646248486705278, 10.03041287825331723519172037823
