| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 3·9-s + 4·11-s − 4·12-s − 4·16-s − 8·17-s + 6·18-s − 4·19-s + 8·22-s + 25-s − 4·27-s − 8·32-s − 8·33-s − 16·34-s + 6·36-s − 8·38-s + 8·41-s + 8·43-s + 8·44-s + 8·48-s − 49-s + 2·50-s + 16·51-s − 8·54-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.20·11-s − 1.15·12-s − 16-s − 1.94·17-s + 1.41·18-s − 0.917·19-s + 1.70·22-s + 1/5·25-s − 0.769·27-s − 1.41·32-s − 1.39·33-s − 2.74·34-s + 36-s − 1.29·38-s + 1.24·41-s + 1.21·43-s + 1.20·44-s + 1.15·48-s − 1/7·49-s + 0.282·50-s + 2.24·51-s − 1.08·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.217504730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.217504730\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.330089320521000015890479908704, −7.60733030562115128588405913803, −7.10737304363177660669565995548, −6.68613128971238834286870447671, −6.35131765939641058418320135022, −6.06203157628456499641950030404, −5.65319593737140052987684536294, −4.91984961860425898516103671255, −4.60125615404385132652450578552, −4.19088060765342639397824093700, −3.90788492159442943708481006165, −3.11986204674958246494658594720, −2.32755441322875807166722683675, −1.81850630484759431090093949383, −0.61550087627193751328171826292,
0.61550087627193751328171826292, 1.81850630484759431090093949383, 2.32755441322875807166722683675, 3.11986204674958246494658594720, 3.90788492159442943708481006165, 4.19088060765342639397824093700, 4.60125615404385132652450578552, 4.91984961860425898516103671255, 5.65319593737140052987684536294, 6.06203157628456499641950030404, 6.35131765939641058418320135022, 6.68613128971238834286870447671, 7.10737304363177660669565995548, 7.60733030562115128588405913803, 8.330089320521000015890479908704
