| L(s) = 1 | + 3-s − 2·7-s + 9-s + 10·13-s + 10·19-s − 2·21-s + 27-s − 2·31-s + 4·37-s + 10·39-s − 2·43-s − 11·49-s + 10·57-s − 26·61-s − 2·63-s + 22·67-s + 4·73-s + 16·79-s + 81-s − 20·91-s − 2·93-s − 14·97-s − 8·103-s − 14·109-s + 4·111-s + 10·117-s + 14·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 2.77·13-s + 2.29·19-s − 0.436·21-s + 0.192·27-s − 0.359·31-s + 0.657·37-s + 1.60·39-s − 0.304·43-s − 1.57·49-s + 1.32·57-s − 3.32·61-s − 0.251·63-s + 2.68·67-s + 0.468·73-s + 1.80·79-s + 1/9·81-s − 2.09·91-s − 0.207·93-s − 1.42·97-s − 0.788·103-s − 1.34·109-s + 0.379·111-s + 0.924·117-s + 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.495130817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.495130817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.108883986383451182036456380552, −8.275919068288280643535776826565, −8.026266654629362330508166642807, −7.72957625844481983843839527768, −6.77234120616327244201676192765, −6.68901441807918237505792173786, −6.03058890823729073510214550931, −5.61334343701912201622834145502, −5.03310337383134036338192767807, −4.24382547629296535183762439094, −3.50841194957311191464253456169, −3.43647697037916365579697730592, −2.81121112269996318424524765777, −1.63891885984744417728938078317, −1.03773521644426615440225296994,
1.03773521644426615440225296994, 1.63891885984744417728938078317, 2.81121112269996318424524765777, 3.43647697037916365579697730592, 3.50841194957311191464253456169, 4.24382547629296535183762439094, 5.03310337383134036338192767807, 5.61334343701912201622834145502, 6.03058890823729073510214550931, 6.68901441807918237505792173786, 6.77234120616327244201676192765, 7.72957625844481983843839527768, 8.026266654629362330508166642807, 8.275919068288280643535776826565, 9.108883986383451182036456380552
