| L(s) = 1 | − 2-s − 4-s + 5-s − 2·7-s + 3·8-s − 10-s − 5·13-s + 2·14-s − 16-s + 17-s − 19-s − 20-s + 23-s − 4·25-s + 5·26-s + 2·28-s − 9·29-s − 8·31-s − 5·32-s − 34-s − 2·35-s − 2·37-s + 38-s + 3·40-s + 3·41-s + 7·43-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s + 1.06·8-s − 0.316·10-s − 1.38·13-s + 0.534·14-s − 1/4·16-s + 0.242·17-s − 0.229·19-s − 0.223·20-s + 0.208·23-s − 4/5·25-s + 0.980·26-s + 0.377·28-s − 1.67·29-s − 1.43·31-s − 0.883·32-s − 0.171·34-s − 0.338·35-s − 0.328·37-s + 0.162·38-s + 0.474·40-s + 0.468·41-s + 1.06·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30729061604743667613050680491, −9.486856844687813433475551610921, −9.232263132127470389713380311071, −7.85873665118377503458386090870, −7.19141577818664657143012339932, −5.87858091946480144424247943451, −4.87958568118636028443679374257, −3.59857441008458632308323977451, −1.99713451755220112161339413052, 0,
1.99713451755220112161339413052, 3.59857441008458632308323977451, 4.87958568118636028443679374257, 5.87858091946480144424247943451, 7.19141577818664657143012339932, 7.85873665118377503458386090870, 9.232263132127470389713380311071, 9.486856844687813433475551610921, 10.30729061604743667613050680491
