| L(s) = 1 | − 3-s + 4-s − 5-s − 7-s − 12-s + 15-s + 16-s + 2·17-s − 19-s − 20-s + 21-s + 27-s − 28-s − 29-s + 35-s − 41-s − 48-s − 2·51-s − 53-s + 57-s + 59-s + 60-s + 64-s + 2·68-s + 2·71-s − 76-s − 79-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 5-s − 7-s − 12-s + 15-s + 16-s + 2·17-s − 19-s − 20-s + 21-s + 27-s − 28-s − 29-s + 35-s − 41-s − 48-s − 2·51-s − 53-s + 57-s + 59-s + 60-s + 64-s + 2·68-s + 2·71-s − 76-s − 79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3770726116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3770726116\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−15.69721150385094496767899354800, −14.63420773928722559629961047382, −12.66665942627658830350838610678, −11.96978648192178724260299450099, −11.09915165564731068328520576306, −10.00684138086693597518137898757, −7.988751469884320746824006089085, −6.74805173897083286463010363905, −5.64723355410045232763599810849, −3.43180500410537741355091128030,
3.43180500410537741355091128030, 5.64723355410045232763599810849, 6.74805173897083286463010363905, 7.988751469884320746824006089085, 10.00684138086693597518137898757, 11.09915165564731068328520576306, 11.96978648192178724260299450099, 12.66665942627658830350838610678, 14.63420773928722559629961047382, 15.69721150385094496767899354800
