| L(s) = 1 | + 4·5-s − 8·11-s + 6·13-s + 12·17-s + 12·19-s + 2·25-s − 12·37-s + 14·49-s − 32·55-s + 8·59-s + 24·65-s + 12·67-s − 12·79-s + 16·83-s + 48·85-s + 48·95-s + 28·103-s + 12·109-s + 36·113-s + 26·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·143-s + 149-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.41·11-s + 1.66·13-s + 2.91·17-s + 2.75·19-s + 2/5·25-s − 1.97·37-s + 2·49-s − 4.31·55-s + 1.04·59-s + 2.97·65-s + 1.46·67-s − 1.35·79-s + 1.75·83-s + 5.20·85-s + 4.92·95-s + 2.75·103-s + 1.14·109-s + 3.38·113-s + 2.36·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.01·143-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.598329991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.598329991\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.494188284496543234592533185470, −8.470998695006067030452834794087, −7.82093092440754144894681407705, −7.68520059205378579792254715144, −7.18110656022317450069979292583, −7.13498184187551402216846705760, −6.10420713325590369903419652340, −5.99439506951107140353563649296, −5.59390660500861806698277016015, −5.50398548892348353576940507334, −5.05695569941686204008980918420, −4.98812013345774031417377351880, −3.79315401965311602918511229232, −3.59415587264762885269762235941, −3.03607875617291664051446808073, −2.97553881573672663816040184424, −2.02023651586568486525854266419, −1.93196424573009023294316374279, −0.980599720423784070927773174783, −0.884067180939445405831851524648,
0.884067180939445405831851524648, 0.980599720423784070927773174783, 1.93196424573009023294316374279, 2.02023651586568486525854266419, 2.97553881573672663816040184424, 3.03607875617291664051446808073, 3.59415587264762885269762235941, 3.79315401965311602918511229232, 4.98812013345774031417377351880, 5.05695569941686204008980918420, 5.50398548892348353576940507334, 5.59390660500861806698277016015, 5.99439506951107140353563649296, 6.10420713325590369903419652340, 7.13498184187551402216846705760, 7.18110656022317450069979292583, 7.68520059205378579792254715144, 7.82093092440754144894681407705, 8.470998695006067030452834794087, 8.494188284496543234592533185470
