| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s − 2·9-s + 10·11-s + 4·14-s + 5·16-s + 4·18-s − 20·22-s − 6·23-s − 9·25-s − 6·28-s − 12·29-s − 6·32-s − 6·36-s − 12·37-s + 14·43-s + 30·44-s + 12·46-s − 3·49-s + 18·50-s + 2·53-s + 8·56-s + 24·58-s + 4·63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s − 2/3·9-s + 3.01·11-s + 1.06·14-s + 5/4·16-s + 0.942·18-s − 4.26·22-s − 1.25·23-s − 9/5·25-s − 1.13·28-s − 2.22·29-s − 1.06·32-s − 36-s − 1.97·37-s + 2.13·43-s + 4.52·44-s + 1.76·46-s − 3/7·49-s + 2.54·50-s + 0.274·53-s + 1.06·56-s + 3.15·58-s + 0.503·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 53 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
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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
−8.516972320011434802360798948761, −7.83480293097670149583575024590, −7.29623394324494884872736782050, −7.03495675051197588547770515922, −6.44081928918084848990950021086, −6.07633542123688337694617503777, −5.86804060339228394057328286514, −5.14152272189939112761012336082, −3.93608084966220470025238364692, −3.73995006998335701262846087394, −3.56257634194984896651241855159, −2.22771868567223326385254473049, −1.97465997072526071206392649552, −1.09230393667940788469276801222, 0,
1.09230393667940788469276801222, 1.97465997072526071206392649552, 2.22771868567223326385254473049, 3.56257634194984896651241855159, 3.73995006998335701262846087394, 3.93608084966220470025238364692, 5.14152272189939112761012336082, 5.86804060339228394057328286514, 6.07633542123688337694617503777, 6.44081928918084848990950021086, 7.03495675051197588547770515922, 7.29623394324494884872736782050, 7.83480293097670149583575024590, 8.516972320011434802360798948761
