L(s) = 1 | + 3-s + 9-s + 11-s − 2·13-s + 2·17-s + 4·19-s + 27-s − 2·29-s + 8·31-s + 33-s + 2·37-s − 2·39-s + 6·41-s − 4·43-s + 8·47-s + 2·51-s + 10·53-s + 4·57-s + 4·59-s + 2·61-s + 12·67-s + 8·71-s − 14·73-s + 8·79-s + 81-s + 12·83-s − 2·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 0.280·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.63·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.868965356\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.868965356\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.51350816683457, −12.22181255339175, −11.74596510222009, −11.36491340459664, −10.72616265257515, −10.24455377164922, −9.789209558568743, −9.489205069057154, −8.995144233633978, −8.442566865408438, −8.009910870454392, −7.580770832586466, −7.086074452134012, −6.699286497480010, −6.036997538912086, −5.522979101674153, −5.073476147766062, −4.461199305055897, −3.959826032872171, −3.481679301108089, −2.847921382930798, −2.434326493419410, −1.843632841882683, −0.9875339306455484, −0.6626145584888935,
0.6626145584888935, 0.9875339306455484, 1.843632841882683, 2.434326493419410, 2.847921382930798, 3.481679301108089, 3.959826032872171, 4.461199305055897, 5.073476147766062, 5.522979101674153, 6.036997538912086, 6.699286497480010, 7.086074452134012, 7.580770832586466, 8.009910870454392, 8.442566865408438, 8.995144233633978, 9.489205069057154, 9.789209558568743, 10.24455377164922, 10.72616265257515, 11.36491340459664, 11.74596510222009, 12.22181255339175, 12.51350816683457