| L(s) = 1 | + 3·2-s + 3·4-s − 3·8-s + 9-s − 2·11-s − 13·16-s + 3·18-s − 6·22-s − 6·23-s + 25-s − 3·29-s − 15·32-s + 3·36-s + 13·37-s − 17·43-s − 6·44-s − 18·46-s + 3·50-s − 15·53-s − 9·58-s + 3·64-s + 5·67-s − 4·71-s − 3·72-s + 39·74-s − 4·79-s − 8·81-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 1.06·8-s + 1/3·9-s − 0.603·11-s − 3.25·16-s + 0.707·18-s − 1.27·22-s − 1.25·23-s + 1/5·25-s − 0.557·29-s − 2.65·32-s + 1/2·36-s + 2.13·37-s − 2.59·43-s − 0.904·44-s − 2.65·46-s + 0.424·50-s − 2.06·53-s − 1.18·58-s + 3/8·64-s + 0.610·67-s − 0.474·71-s − 0.353·72-s + 4.53·74-s − 0.450·79-s − 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304927 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304927 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 127 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 15 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 93 T^{2} + p^{2} T^{4} \) |
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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
−8.376235919645744588680498612985, −8.146768063952007773629782046787, −7.60618368093220024237124522100, −6.89694339732238717200344841029, −6.29774301985203867958611827490, −6.16934913023246681005189407165, −5.41101634315997015782412637012, −5.17967285299293180914106695486, −4.60471145893697746959349398858, −4.16172541632434519708907903166, −3.76416890388226111834559063911, −3.05376541926548206386059242744, −2.66778158053672176257314357951, −1.69604112364983410564776452035, 0,
1.69604112364983410564776452035, 2.66778158053672176257314357951, 3.05376541926548206386059242744, 3.76416890388226111834559063911, 4.16172541632434519708907903166, 4.60471145893697746959349398858, 5.17967285299293180914106695486, 5.41101634315997015782412637012, 6.16934913023246681005189407165, 6.29774301985203867958611827490, 6.89694339732238717200344841029, 7.60618368093220024237124522100, 8.146768063952007773629782046787, 8.376235919645744588680498612985
