| L(s) = 1 | + 1.90·2-s + 1.61·4-s + 1.61·5-s − 1.61·7-s − 0.726·8-s + 3.07·10-s − 0.236·11-s − 4.97·13-s − 3.07·14-s − 4.61·16-s + 0.236·17-s + 2.61·20-s − 0.449·22-s − 8.23·23-s − 2.38·25-s − 9.47·26-s − 2.61·28-s − 8.50·29-s + 7.33·31-s − 7.33·32-s + 0.449·34-s − 2.61·35-s + 5.25·37-s − 1.17·40-s + 3.52·41-s − 8.23·43-s − 0.381·44-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.809·4-s + 0.723·5-s − 0.611·7-s − 0.256·8-s + 0.973·10-s − 0.0711·11-s − 1.38·13-s − 0.822·14-s − 1.15·16-s + 0.0572·17-s + 0.585·20-s − 0.0957·22-s − 1.71·23-s − 0.476·25-s − 1.85·26-s − 0.494·28-s − 1.57·29-s + 1.31·31-s − 1.29·32-s + 0.0770·34-s − 0.442·35-s + 0.864·37-s − 0.185·40-s + 0.550·41-s − 1.25·43-s − 0.0575·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 - 7.33T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 8.95T + 67T^{2} \) |
| 71 | \( 1 + 8.78T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 + 0.171T + 79T^{2} \) |
| 83 | \( 1 + 9.94T + 83T^{2} \) |
| 89 | \( 1 + 0.898T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.123013330595770101773611989253, −7.32039465485640576984578427190, −6.38850838569768655240696643787, −5.93086796828480236078014638798, −5.22029375580666842431987073646, −4.41942033865901200295153595445, −3.65543650845694511113691149472, −2.67425313692578879012607492365, −2.02250705905957479353052075096, 0,
2.02250705905957479353052075096, 2.67425313692578879012607492365, 3.65543650845694511113691149472, 4.41942033865901200295153595445, 5.22029375580666842431987073646, 5.93086796828480236078014638798, 6.38850838569768655240696643787, 7.32039465485640576984578427190, 8.123013330595770101773611989253
