| L(s) = 1 | − 4-s + 2·9-s − 2·11-s + 16-s − 4·19-s + 16·31-s − 2·36-s + 2·44-s − 49-s − 24·59-s − 20·61-s − 64-s − 24·71-s + 4·76-s + 20·79-s − 5·81-s + 36·89-s − 4·99-s − 36·101-s − 16·109-s + 3·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s − 0.603·11-s + 1/4·16-s − 0.917·19-s + 2.87·31-s − 1/3·36-s + 0.301·44-s − 1/7·49-s − 3.12·59-s − 2.56·61-s − 1/8·64-s − 2.84·71-s + 0.458·76-s + 2.25·79-s − 5/9·81-s + 3.81·89-s − 0.402·99-s − 3.58·101-s − 1.53·109-s + 3/11·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.116365740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116365740\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.042254935673947115081976576770, −8.110100837725497224495687874140, −7.897894457760361382740339582380, −7.81840832332704904083289543596, −7.29935467567606239808091724305, −6.64528148177060704380301008250, −6.53698197148980954081587392981, −6.03739051632072380422913906033, −5.87955401896915691480635373505, −5.12961404772351430051208950385, −4.71980815172921511580929961786, −4.62626840556589829670061105187, −4.21755996169935317041061406748, −3.73134471099496489293692804294, −3.03290077045981257801998467595, −2.87810699699757592728356600876, −2.31580726865117764745013998178, −1.54959607914988530110163358505, −1.25194369111952391920379542501, −0.32414433511706575374543522213,
0.32414433511706575374543522213, 1.25194369111952391920379542501, 1.54959607914988530110163358505, 2.31580726865117764745013998178, 2.87810699699757592728356600876, 3.03290077045981257801998467595, 3.73134471099496489293692804294, 4.21755996169935317041061406748, 4.62626840556589829670061105187, 4.71980815172921511580929961786, 5.12961404772351430051208950385, 5.87955401896915691480635373505, 6.03739051632072380422913906033, 6.53698197148980954081587392981, 6.64528148177060704380301008250, 7.29935467567606239808091724305, 7.81840832332704904083289543596, 7.897894457760361382740339582380, 8.110100837725497224495687874140, 9.042254935673947115081976576770
