L(s) = 1 | + 2·5-s − 11-s − 5·13-s + 4·17-s + 19-s − 2·23-s − 25-s + 2·29-s + 31-s + 5·37-s + 3·43-s − 4·53-s − 2·55-s + 2·59-s − 10·61-s − 10·65-s + 11·67-s − 10·71-s − 11·73-s − 5·79-s − 18·83-s + 8·85-s + 18·89-s + 2·95-s − 14·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.301·11-s − 1.38·13-s + 0.970·17-s + 0.229·19-s − 0.417·23-s − 1/5·25-s + 0.371·29-s + 0.179·31-s + 0.821·37-s + 0.457·43-s − 0.549·53-s − 0.269·55-s + 0.260·59-s − 1.28·61-s − 1.24·65-s + 1.34·67-s − 1.18·71-s − 1.28·73-s − 0.562·79-s − 1.97·83-s + 0.867·85-s + 1.90·89-s + 0.205·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.031775475\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.031775475\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.72708891207557, −12.15952995301123, −11.84218545235028, −11.37741710045284, −10.65922065457200, −10.28991436249130, −9.853942966077799, −9.593559323279679, −9.156527870604443, −8.469408922338622, −7.982612417458745, −7.511831153892249, −7.183793407660610, −6.516583343267819, −5.934855526623911, −5.670019555629256, −5.131452225266931, −4.595854624854079, −4.158159653103441, −3.341342265074555, −2.799865824661584, −2.432470616216674, −1.757049356491730, −1.218667118447266, −0.3723505059819601,
0.3723505059819601, 1.218667118447266, 1.757049356491730, 2.432470616216674, 2.799865824661584, 3.341342265074555, 4.158159653103441, 4.595854624854079, 5.131452225266931, 5.670019555629256, 5.934855526623911, 6.516583343267819, 7.183793407660610, 7.511831153892249, 7.982612417458745, 8.469408922338622, 9.156527870604443, 9.593559323279679, 9.853942966077799, 10.28991436249130, 10.65922065457200, 11.37741710045284, 11.84218545235028, 12.15952995301123, 12.72708891207557