| L(s) = 1 | − 18·5-s − 7·7-s − 72·11-s − 34·13-s − 6·17-s − 92·19-s − 180·23-s + 199·25-s + 114·29-s − 56·31-s + 126·35-s − 34·37-s − 6·41-s − 164·43-s + 168·47-s + 49·49-s − 654·53-s + 1.29e3·55-s − 492·59-s − 250·61-s + 612·65-s + 124·67-s + 36·71-s + 1.01e3·73-s + 504·77-s − 56·79-s + 228·83-s + ⋯ |
| L(s) = 1 | − 1.60·5-s − 0.377·7-s − 1.97·11-s − 0.725·13-s − 0.0856·17-s − 1.11·19-s − 1.63·23-s + 1.59·25-s + 0.729·29-s − 0.324·31-s + 0.608·35-s − 0.151·37-s − 0.0228·41-s − 0.581·43-s + 0.521·47-s + 1/7·49-s − 1.69·53-s + 3.17·55-s − 1.08·59-s − 0.524·61-s + 1.16·65-s + 0.226·67-s + 0.0601·71-s + 1.61·73-s + 0.745·77-s − 0.0797·79-s + 0.301·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1211307449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1211307449\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 72 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 180 T + p^{3} T^{2} \) |
| 29 | \( 1 - 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 654 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 - 124 T + p^{3} T^{2} \) |
| 71 | \( 1 - 36 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 + 56 T + p^{3} T^{2} \) |
| 83 | \( 1 - 228 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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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
−9.695405985698801361712689478116, −8.386927936121197134401679632456, −7.972419849472358323371903806287, −7.31228432294166954801194282258, −6.24409825871874111606320585730, −5.04154671996810302421728666969, −4.30023759576056871839957999833, −3.26887298213722155725537289624, −2.29494024570458490536109231618, −0.17089014794281650893525064540,
0.17089014794281650893525064540, 2.29494024570458490536109231618, 3.26887298213722155725537289624, 4.30023759576056871839957999833, 5.04154671996810302421728666969, 6.24409825871874111606320585730, 7.31228432294166954801194282258, 7.972419849472358323371903806287, 8.386927936121197134401679632456, 9.695405985698801361712689478116
