| L(s) = 1 | + 3-s − 4-s + 2·7-s − 2·9-s − 3·11-s − 12-s + 16-s + 2·21-s − 2·25-s − 5·27-s − 2·28-s − 3·33-s + 2·36-s + 2·37-s + 3·44-s + 6·47-s + 48-s − 2·49-s − 18·53-s − 4·63-s − 64-s + 5·67-s + 5·73-s − 2·75-s − 6·77-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 0.755·7-s − 2/3·9-s − 0.904·11-s − 0.288·12-s + 1/4·16-s + 0.436·21-s − 2/5·25-s − 0.962·27-s − 0.377·28-s − 0.522·33-s + 1/3·36-s + 0.328·37-s + 0.452·44-s + 0.875·47-s + 0.144·48-s − 2/7·49-s − 2.47·53-s − 0.503·63-s − 1/8·64-s + 0.610·67-s + 0.585·73-s − 0.230·75-s − 0.683·77-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
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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
−8.221693457239718930678969743346, −8.078619767205698695728525082814, −7.59210974849329653889914934402, −7.25465632962485148090824094713, −6.29961927548681697410739213638, −6.11822534305357046284682785556, −5.40150976461810021290912525157, −4.94960844967772864236710814837, −4.68644294709606454566545627945, −3.79719968254017802650052862729, −3.47645645180002197517998353982, −2.64663352295927200383301861618, −2.25366081924076436626125130117, −1.31666493538200555722226499369, 0,
1.31666493538200555722226499369, 2.25366081924076436626125130117, 2.64663352295927200383301861618, 3.47645645180002197517998353982, 3.79719968254017802650052862729, 4.68644294709606454566545627945, 4.94960844967772864236710814837, 5.40150976461810021290912525157, 6.11822534305357046284682785556, 6.29961927548681697410739213638, 7.25465632962485148090824094713, 7.59210974849329653889914934402, 8.078619767205698695728525082814, 8.221693457239718930678969743346
