| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 9-s − 2·10-s + 2·12-s + 15-s − 4·16-s − 2·18-s + 2·20-s + 25-s + 27-s − 2·30-s − 20·31-s + 8·32-s + 2·36-s + 45-s − 4·48-s − 10·49-s − 2·50-s − 20·53-s − 2·54-s + 2·60-s + 40·62-s − 8·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.577·12-s + 0.258·15-s − 16-s − 0.471·18-s + 0.447·20-s + 1/5·25-s + 0.192·27-s − 0.365·30-s − 3.59·31-s + 1.41·32-s + 1/3·36-s + 0.149·45-s − 0.577·48-s − 1.42·49-s − 0.282·50-s − 2.74·53-s − 0.272·54-s + 0.258·60-s + 5.08·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.777463202787256456016698078145, −8.568121348401068092389681947177, −7.85126782769779844678487827092, −7.55865663172740494140791662232, −7.14925418958104587435907867580, −6.60444979582363780589168413242, −6.03848785485222247521986180613, −5.39414327193829845230627529171, −4.83299639814930091125721935543, −4.14575433902751758600957781044, −3.43838310204251765276920180681, −2.80479014504434913744491313053, −1.80502156484415218287852213649, −1.59243496351402759769462736519, 0,
1.59243496351402759769462736519, 1.80502156484415218287852213649, 2.80479014504434913744491313053, 3.43838310204251765276920180681, 4.14575433902751758600957781044, 4.83299639814930091125721935543, 5.39414327193829845230627529171, 6.03848785485222247521986180613, 6.60444979582363780589168413242, 7.14925418958104587435907867580, 7.55865663172740494140791662232, 7.85126782769779844678487827092, 8.568121348401068092389681947177, 8.777463202787256456016698078145
