| L(s) = 1 | + 13·7-s + 13-s + 11·19-s + 25·25-s + 46·31-s − 47·37-s − 22·43-s + 120·49-s + 121·61-s − 109·67-s − 97·73-s − 131·79-s + 13·91-s + 167·97-s + 37·103-s + 214·109-s + ⋯ |
| L(s) = 1 | + 13/7·7-s + 1/13·13-s + 0.578·19-s + 25-s + 1.48·31-s − 1.27·37-s − 0.511·43-s + 2.44·49-s + 1.98·61-s − 1.62·67-s − 1.32·73-s − 1.65·79-s + 1/7·91-s + 1.72·97-s + 0.359·103-s + 1.96·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.765955469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.765955469\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 13 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 46 T + p^{2} T^{2} \) |
| 37 | \( 1 + 47 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 121 T + p^{2} T^{2} \) |
| 67 | \( 1 + 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 97 T + p^{2} T^{2} \) |
| 79 | \( 1 + 131 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 167 T + p^{2} T^{2} \) |
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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
−8.832191132716513560023163566445, −8.441424855054702279107561522926, −7.61170707879741693212789706556, −6.90276835130620902396001436765, −5.73918115331719662954189664355, −4.95030678163058239940814483545, −4.37035327316979144224319786877, −3.09540662690839781092232641970, −1.92046555309253659993494325977, −0.990493915382002457614950860322,
0.990493915382002457614950860322, 1.92046555309253659993494325977, 3.09540662690839781092232641970, 4.37035327316979144224319786877, 4.95030678163058239940814483545, 5.73918115331719662954189664355, 6.90276835130620902396001436765, 7.61170707879741693212789706556, 8.441424855054702279107561522926, 8.832191132716513560023163566445
