| L(s) = 1 | − 2·4-s − 3·5-s − 7-s − 11-s + 4·16-s + 2·17-s + 6·20-s − 4·23-s + 4·25-s + 2·28-s − 5·31-s + 3·35-s + 10·37-s + 5·41-s + 4·43-s + 2·44-s + 3·47-s + 49-s + 5·53-s + 3·55-s − 10·59-s − 5·61-s − 8·64-s − 10·67-s − 4·68-s + 15·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s − 0.377·7-s − 0.301·11-s + 16-s + 0.485·17-s + 1.34·20-s − 0.834·23-s + 4/5·25-s + 0.377·28-s − 0.898·31-s + 0.507·35-s + 1.64·37-s + 0.780·41-s + 0.609·43-s + 0.301·44-s + 0.437·47-s + 1/7·49-s + 0.686·53-s + 0.404·55-s − 1.30·59-s − 0.640·61-s − 64-s − 1.22·67-s − 0.485·68-s + 1.78·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−13.00398318231597, −12.54701362593836, −12.25034956823943, −11.85746945661913, −11.13262465056810, −10.89148230508747, −10.28065335573305, −9.760953482248170, −9.306855217677972, −8.970905297118879, −8.228205344153013, −8.005256678785357, −7.515661460096959, −7.229600325045549, −6.298023469272753, −5.925644731521183, −5.378620099853290, −4.763810204177743, −4.234187728960620, −3.942135553137627, −3.426796207884598, −2.879945696950646, −2.177727246703537, −1.194760801558737, −0.5962213587445697, 0,
0.5962213587445697, 1.194760801558737, 2.177727246703537, 2.879945696950646, 3.426796207884598, 3.942135553137627, 4.234187728960620, 4.763810204177743, 5.378620099853290, 5.925644731521183, 6.298023469272753, 7.229600325045549, 7.515661460096959, 8.005256678785357, 8.228205344153013, 8.970905297118879, 9.306855217677972, 9.760953482248170, 10.28065335573305, 10.89148230508747, 11.13262465056810, 11.85746945661913, 12.25034956823943, 12.54701362593836, 13.00398318231597
