| L(s) = 1 | − 2·3-s − 4·4-s − 3·9-s − 3·11-s + 8·12-s + 12·16-s + 12·23-s + 14·27-s − 8·31-s + 6·33-s + 12·36-s − 4·37-s + 12·44-s − 18·47-s − 24·48-s + 49-s − 24·53-s − 32·64-s + 8·67-s − 24·69-s − 4·81-s − 24·89-s − 48·92-s + 16·93-s + 2·97-s + 9·99-s − 10·103-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s − 9-s − 0.904·11-s + 2.30·12-s + 3·16-s + 2.50·23-s + 2.69·27-s − 1.43·31-s + 1.04·33-s + 2·36-s − 0.657·37-s + 1.80·44-s − 2.62·47-s − 3.46·48-s + 1/7·49-s − 3.29·53-s − 4·64-s + 0.977·67-s − 2.88·69-s − 4/9·81-s − 2.54·89-s − 5.00·92-s + 1.65·93-s + 0.203·97-s + 0.904·99-s − 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3705625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3705625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−6.93454011637102289378074371468, −6.53425194229678634962047335445, −6.09018313766691468884559494421, −5.45020178597402395728815947142, −5.36247556826240082732363502623, −5.03402919049841470357200611331, −4.78686073935120916368506592009, −4.25025415642659253528011766319, −3.49570769800642512352485217195, −3.00779173186661481858377770522, −2.98704088538274633622207409794, −1.67004050212900524489803650478, −0.967342094538072069158993731603, 0, 0,
0.967342094538072069158993731603, 1.67004050212900524489803650478, 2.98704088538274633622207409794, 3.00779173186661481858377770522, 3.49570769800642512352485217195, 4.25025415642659253528011766319, 4.78686073935120916368506592009, 5.03402919049841470357200611331, 5.36247556826240082732363502623, 5.45020178597402395728815947142, 6.09018313766691468884559494421, 6.53425194229678634962047335445, 6.93454011637102289378074371468
