| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 7-s − 2.23·8-s + 5.61·11-s + 4.23·13-s + 0.618·14-s + 1.85·16-s + 0.618·17-s − 1.85·19-s + 3.47·22-s + 1.47·23-s + 2.61·26-s − 1.61·28-s + 5.38·29-s + 6.70·31-s + 5.61·32-s + 0.381·34-s − 37-s − 1.14·38-s + 0.527·41-s + 11.5·43-s − 9.09·44-s + 0.909·46-s − 1.47·47-s − 6·49-s − 6.85·52-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.377·7-s − 0.790·8-s + 1.69·11-s + 1.17·13-s + 0.165·14-s + 0.463·16-s + 0.149·17-s − 0.425·19-s + 0.740·22-s + 0.306·23-s + 0.513·26-s − 0.305·28-s + 0.999·29-s + 1.20·31-s + 0.993·32-s + 0.0655·34-s − 0.164·37-s − 0.185·38-s + 0.0824·41-s + 1.76·43-s − 1.37·44-s + 0.134·46-s − 0.214·47-s − 0.857·49-s − 0.950·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.697139987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.697139987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 41 | \( 1 - 0.527T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 4.32T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 - 6.09T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−7.983625088316655646693783743799, −6.95089036993319066649407894078, −6.15631548986346549761384845687, −5.93571029277562812804595364806, −4.68223073241857679996306695260, −4.41752499057238022810604101796, −3.62590004640087670715908122838, −2.95118127778411766854116589824, −1.55445804651121496678999904601, −0.843344370285554550228455898524,
0.843344370285554550228455898524, 1.55445804651121496678999904601, 2.95118127778411766854116589824, 3.62590004640087670715908122838, 4.41752499057238022810604101796, 4.68223073241857679996306695260, 5.93571029277562812804595364806, 6.15631548986346549761384845687, 6.95089036993319066649407894078, 7.983625088316655646693783743799
