| L(s) = 1 | − 4·5-s − 9-s + 12·11-s + 12·19-s + 11·25-s − 4·29-s − 8·31-s − 12·41-s + 4·45-s + 14·49-s − 48·55-s + 20·59-s − 12·61-s + 16·71-s + 32·79-s + 81-s + 20·89-s − 48·95-s − 12·99-s − 4·101-s + 32·109-s + 86·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1/3·9-s + 3.61·11-s + 2.75·19-s + 11/5·25-s − 0.742·29-s − 1.43·31-s − 1.87·41-s + 0.596·45-s + 2·49-s − 6.47·55-s + 2.60·59-s − 1.53·61-s + 1.89·71-s + 3.60·79-s + 1/9·81-s + 2.11·89-s − 4.92·95-s − 1.20·99-s − 0.398·101-s + 3.06·109-s + 7.81·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.947482231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.947482231\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.985638146542112474649593913011, −8.605584124902338579934079306047, −8.014898246931842699034920107393, −7.79859561663993858704704282312, −7.22072733554941409889066020828, −7.14499278852233543646786149229, −6.65812003953942996506590022210, −6.53827359966015120883962012832, −5.68929262885963082361920206186, −5.58868514554132514169985622503, −4.79248334589845639249736729994, −4.72049491964891259502320258345, −3.81001448163289466907465273624, −3.71383187391819415176085342724, −3.56158912243356156542751275670, −3.27989569536205576237731261848, −2.24150652283771042733532585058, −1.65176243207974130759187324214, −0.905334442637417059981850343670, −0.76724170859269516958330879521,
0.76724170859269516958330879521, 0.905334442637417059981850343670, 1.65176243207974130759187324214, 2.24150652283771042733532585058, 3.27989569536205576237731261848, 3.56158912243356156542751275670, 3.71383187391819415176085342724, 3.81001448163289466907465273624, 4.72049491964891259502320258345, 4.79248334589845639249736729994, 5.58868514554132514169985622503, 5.68929262885963082361920206186, 6.53827359966015120883962012832, 6.65812003953942996506590022210, 7.14499278852233543646786149229, 7.22072733554941409889066020828, 7.79859561663993858704704282312, 8.014898246931842699034920107393, 8.605584124902338579934079306047, 8.985638146542112474649593913011
