| L(s) = 1 | + 2·3-s + 4-s + 2·5-s + 3·9-s − 4·11-s + 2·12-s + 4·15-s + 16-s + 2·20-s − 16·23-s + 3·25-s + 4·27-s − 8·33-s + 3·36-s + 12·37-s − 4·44-s + 6·45-s + 2·48-s + 49-s − 20·53-s − 8·55-s + 24·59-s + 4·60-s + 64-s − 24·67-s − 32·69-s − 16·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s − 1.20·11-s + 0.577·12-s + 1.03·15-s + 1/4·16-s + 0.447·20-s − 3.33·23-s + 3/5·25-s + 0.769·27-s − 1.39·33-s + 1/2·36-s + 1.97·37-s − 0.603·44-s + 0.894·45-s + 0.288·48-s + 1/7·49-s − 2.74·53-s − 1.07·55-s + 3.12·59-s + 0.516·60-s + 1/8·64-s − 2.93·67-s − 3.85·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5336100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5336100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.31970008287706573052341981271, −6.66448553069377372254046606652, −6.09618123486281782132752637194, −5.97975998469911468351052676550, −5.70040506753513954559543286636, −4.95725936124254114017607791343, −4.44741136509767419706316851812, −4.25799725591787504630789976261, −3.44893990823261081236809071688, −3.20504127426248803147211761419, −2.48074564582371685735453886133, −2.27051973415844153304852765225, −1.89313461948809196331835209034, −1.19897415920148196135153136845, 0,
1.19897415920148196135153136845, 1.89313461948809196331835209034, 2.27051973415844153304852765225, 2.48074564582371685735453886133, 3.20504127426248803147211761419, 3.44893990823261081236809071688, 4.25799725591787504630789976261, 4.44741136509767419706316851812, 4.95725936124254114017607791343, 5.70040506753513954559543286636, 5.97975998469911468351052676550, 6.09618123486281782132752637194, 6.66448553069377372254046606652, 7.31970008287706573052341981271
