| L(s) = 1 | + 2·3-s + 3·9-s + 4·17-s − 8·19-s − 6·25-s + 4·27-s − 12·41-s − 16·43-s + 49-s + 8·51-s − 16·57-s + 24·59-s − 32·67-s − 12·73-s − 12·75-s + 5·81-s + 24·83-s − 28·89-s − 12·97-s + 16·107-s + 4·113-s − 22·121-s − 24·123-s + 127-s − 32·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.970·17-s − 1.83·19-s − 6/5·25-s + 0.769·27-s − 1.87·41-s − 2.43·43-s + 1/7·49-s + 1.12·51-s − 2.11·57-s + 3.12·59-s − 3.90·67-s − 1.40·73-s − 1.38·75-s + 5/9·81-s + 2.63·83-s − 2.96·89-s − 1.21·97-s + 1.54·107-s + 0.376·113-s − 2·121-s − 2.16·123-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.166047761688637687090227543209, −7.59021173617066708905907369573, −7.00761877497144406942618978881, −6.91666929474613726367436300689, −6.06997282501662098824886849744, −5.92091437501965485968615955402, −5.03856124527433249174894736338, −4.78854214325172504653136346628, −3.89046750453116338353014311196, −3.86717148280360085793211429715, −3.13298221583067644067427770880, −2.62233033257522210046559530213, −1.87613834788093784477265186390, −1.50798537152368882588814373343, 0,
1.50798537152368882588814373343, 1.87613834788093784477265186390, 2.62233033257522210046559530213, 3.13298221583067644067427770880, 3.86717148280360085793211429715, 3.89046750453116338353014311196, 4.78854214325172504653136346628, 5.03856124527433249174894736338, 5.92091437501965485968615955402, 6.06997282501662098824886849744, 6.91666929474613726367436300689, 7.00761877497144406942618978881, 7.59021173617066708905907369573, 8.166047761688637687090227543209
