| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s + 7-s − 8-s − 10-s − 2·13-s + 14-s − 15-s − 16-s + 2·19-s + 21-s − 24-s + 25-s − 2·26-s − 27-s − 29-s − 30-s − 35-s + 2·38-s − 2·39-s + 40-s + 42-s + 43-s − 48-s + 50-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s + 7-s − 8-s − 10-s − 2·13-s + 14-s − 15-s − 16-s + 2·19-s + 21-s − 24-s + 25-s − 2·26-s − 27-s − 29-s − 30-s − 35-s + 2·38-s − 2·39-s + 40-s + 42-s + 43-s − 48-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.307408824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307408824\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−11.92212510875470901497363276412, −11.32316503911993598348756430797, −9.706073934590775688607269601835, −8.932206368135669341641465712365, −7.81378053393453958063549501241, −7.35605807477426326357152382534, −5.41401075361990551667687990946, −4.69273044343461796129640778052, −3.58458439254698469851743517526, −2.59892916973020532017446207408,
2.59892916973020532017446207408, 3.58458439254698469851743517526, 4.69273044343461796129640778052, 5.41401075361990551667687990946, 7.35605807477426326357152382534, 7.81378053393453958063549501241, 8.932206368135669341641465712365, 9.706073934590775688607269601835, 11.32316503911993598348756430797, 11.92212510875470901497363276412
