| L(s) = 1 | + 2-s − 4-s + 2·5-s + 2·7-s − 3·8-s − 3·9-s + 2·10-s − 2·11-s − 6·13-s + 2·14-s − 16-s + 2·17-s − 3·18-s + 8·19-s − 2·20-s − 2·22-s + 4·23-s − 25-s − 6·26-s − 2·28-s + 2·29-s − 2·31-s + 5·32-s + 2·34-s + 4·35-s + 3·36-s − 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 1.06·8-s − 9-s + 0.632·10-s − 0.603·11-s − 1.66·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.83·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s − 1/5·25-s − 1.17·26-s − 0.377·28-s + 0.371·29-s − 0.359·31-s + 0.883·32-s + 0.342·34-s + 0.676·35-s + 1/2·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182660467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182660467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.20095311456821689571194047713, −13.90079370363007228028454045175, −12.55583978793064302952805910689, −11.60505678784583813722793889342, −10.02750085781246502621897747270, −9.047241577107533133970814126334, −7.59174437898385961503046561534, −5.58836450574614755457010078237, −5.03053356685895702308389540370, −2.85172594127170398516699652084,
2.85172594127170398516699652084, 5.03053356685895702308389540370, 5.58836450574614755457010078237, 7.59174437898385961503046561534, 9.047241577107533133970814126334, 10.02750085781246502621897747270, 11.60505678784583813722793889342, 12.55583978793064302952805910689, 13.90079370363007228028454045175, 14.20095311456821689571194047713
