| L(s) = 1 | + 3-s + 9-s − 12·11-s + 4·13-s − 12·23-s − 10·25-s + 27-s − 12·33-s + 4·37-s + 4·39-s + 24·47-s + 49-s − 20·61-s − 12·69-s + 12·71-s − 20·73-s − 10·75-s + 81-s − 24·83-s − 20·97-s − 12·99-s − 12·107-s + 28·109-s + 4·111-s + 4·117-s + 86·121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 3.61·11-s + 1.10·13-s − 2.50·23-s − 2·25-s + 0.192·27-s − 2.08·33-s + 0.657·37-s + 0.640·39-s + 3.50·47-s + 1/7·49-s − 2.56·61-s − 1.44·69-s + 1.42·71-s − 2.34·73-s − 1.15·75-s + 1/9·81-s − 2.63·83-s − 2.03·97-s − 1.20·99-s − 1.16·107-s + 2.68·109-s + 0.379·111-s + 0.369·117-s + 7.81·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.666722228038389417268306810761, −8.701525582792892098188512767049, −8.418500928498397092746321719715, −7.76051152200915202719683008376, −7.74489471466623660805940258686, −7.26031081711332154770334058409, −5.98540521798352690518559815387, −5.82277620068151436147210793072, −5.48036643270637995428248023009, −4.39050801784138122129668187119, −4.11399937114205741213147454042, −3.12875372570045370376261059932, −2.51752502799515976298852414682, −1.96257743518867342978553596046, 0,
1.96257743518867342978553596046, 2.51752502799515976298852414682, 3.12875372570045370376261059932, 4.11399937114205741213147454042, 4.39050801784138122129668187119, 5.48036643270637995428248023009, 5.82277620068151436147210793072, 5.98540521798352690518559815387, 7.26031081711332154770334058409, 7.74489471466623660805940258686, 7.76051152200915202719683008376, 8.418500928498397092746321719715, 8.701525582792892098188512767049, 9.666722228038389417268306810761
