L(s) = 1 | − 5-s + 4·11-s + 13-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s + 8·31-s − 10·37-s + 6·41-s − 4·43-s − 7·49-s + 10·53-s − 4·55-s + 4·59-s − 2·61-s − 65-s + 12·67-s + 16·71-s + 2·73-s + 16·79-s − 12·83-s − 6·85-s − 10·89-s + 4·95-s − 6·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s + 0.277·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.937·41-s − 0.609·43-s − 49-s + 1.37·53-s − 0.539·55-s + 0.520·59-s − 0.256·61-s − 0.124·65-s + 1.46·67-s + 1.89·71-s + 0.234·73-s + 1.80·79-s − 1.31·83-s − 0.650·85-s − 1.05·89-s + 0.410·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.263013966\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.263013966\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77880647931302564592904046027, −6.75265964204349221579119142608, −6.68865403873865633954331339527, −5.55396349254493865227638678725, −5.02063230332142551732978949848, −3.99963285373042159663463440440, −3.61006721193538909528925846668, −2.72534016693095438477991612325, −1.56263206681440077914997356281, −0.77659600232925668982734274907,
0.77659600232925668982734274907, 1.56263206681440077914997356281, 2.72534016693095438477991612325, 3.61006721193538909528925846668, 3.99963285373042159663463440440, 5.02063230332142551732978949848, 5.55396349254493865227638678725, 6.68865403873865633954331339527, 6.75265964204349221579119142608, 7.77880647931302564592904046027