| L(s) = 1 | − 2·5-s + 2·11-s + 6·19-s − 25-s + 6·29-s + 2·31-s + 2·41-s + 10·49-s − 4·55-s + 26·59-s − 20·61-s + 18·71-s − 26·89-s − 12·95-s − 2·101-s + 34·109-s − 19·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s − 4·155-s + 157-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s + 1.37·19-s − 1/5·25-s + 1.11·29-s + 0.359·31-s + 0.312·41-s + 10/7·49-s − 0.539·55-s + 3.38·59-s − 2.56·61-s + 2.13·71-s − 2.75·89-s − 1.23·95-s − 0.199·101-s + 3.25·109-s − 1.72·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s − 0.321·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084114111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084114111\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.752663051235749526247372434375, −9.156776869237081681227048108457, −8.638053333556294759308635496699, −8.545600757907975704833084053319, −7.80934416010358008910074318895, −7.80141195827052797798254659887, −7.14745750364661776362062473862, −6.89022073050100050132739702633, −6.49583072588978968921695620220, −5.86716026524831793682116781716, −5.51059894091704824499375695885, −5.10965468354132988261159119786, −4.32295660090599444010286213010, −4.31476067243051768989509818281, −3.61855226745933147483145720661, −3.25145784805764919485272193723, −2.68154440202702238267960927533, −2.06085513851116566833499558608, −1.17352379672611871956514233766, −0.65003481834077567201944889585,
0.65003481834077567201944889585, 1.17352379672611871956514233766, 2.06085513851116566833499558608, 2.68154440202702238267960927533, 3.25145784805764919485272193723, 3.61855226745933147483145720661, 4.31476067243051768989509818281, 4.32295660090599444010286213010, 5.10965468354132988261159119786, 5.51059894091704824499375695885, 5.86716026524831793682116781716, 6.49583072588978968921695620220, 6.89022073050100050132739702633, 7.14745750364661776362062473862, 7.80141195827052797798254659887, 7.80934416010358008910074318895, 8.545600757907975704833084053319, 8.638053333556294759308635496699, 9.156776869237081681227048108457, 9.752663051235749526247372434375
