| L(s) = 1 | + 2·3-s + 2·5-s + 2·9-s − 4·11-s + 6·13-s + 4·15-s − 2·19-s − 8·23-s − 2·25-s + 6·27-s + 4·31-s − 8·33-s + 12·39-s + 8·41-s + 4·43-s + 4·45-s + 12·47-s + 20·53-s − 8·55-s − 4·57-s + 14·59-s + 18·61-s + 12·65-s + 8·67-s − 16·69-s − 8·71-s − 12·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 2/3·9-s − 1.20·11-s + 1.66·13-s + 1.03·15-s − 0.458·19-s − 1.66·23-s − 2/5·25-s + 1.15·27-s + 0.718·31-s − 1.39·33-s + 1.92·39-s + 1.24·41-s + 0.609·43-s + 0.596·45-s + 1.75·47-s + 2.74·53-s − 1.07·55-s − 0.529·57-s + 1.82·59-s + 2.30·61-s + 1.48·65-s + 0.977·67-s − 1.92·69-s − 0.949·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.479062298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.479062298\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.720158930882663131053464858152, −8.514538596578400562671092891999, −8.185107943971822856923139347137, −7.994647389976667672137548719805, −7.45051568835486467965871940700, −7.03468266134560376851255626323, −6.66366136447287891674364888926, −6.17319485261003624198541789241, −5.73720179277806528353072995806, −5.58839402132175013148296073618, −5.24099471706668526143146906333, −4.27226927887734482310772203548, −4.17184093824631510286201721847, −3.82588966896805119368913868623, −3.26063716956948923501435830966, −2.52385360263673289479409211698, −2.40335105737748243788821728374, −2.12933544039847450994978797342, −1.23663477295016317268001549261, −0.70969926236305266170088297343,
0.70969926236305266170088297343, 1.23663477295016317268001549261, 2.12933544039847450994978797342, 2.40335105737748243788821728374, 2.52385360263673289479409211698, 3.26063716956948923501435830966, 3.82588966896805119368913868623, 4.17184093824631510286201721847, 4.27226927887734482310772203548, 5.24099471706668526143146906333, 5.58839402132175013148296073618, 5.73720179277806528353072995806, 6.17319485261003624198541789241, 6.66366136447287891674364888926, 7.03468266134560376851255626323, 7.45051568835486467965871940700, 7.994647389976667672137548719805, 8.185107943971822856923139347137, 8.514538596578400562671092891999, 8.720158930882663131053464858152
