| L(s) = 1 | + 15·5-s − 25·7-s − 15·11-s − 20·13-s − 72·17-s − 2·19-s − 114·23-s + 100·25-s + 30·29-s + 101·31-s − 375·35-s + 430·37-s + 30·41-s − 110·43-s + 330·47-s + 282·49-s + 621·53-s − 225·55-s − 660·59-s + 376·61-s − 300·65-s + 250·67-s + 360·71-s + 785·73-s + 375·77-s + 488·79-s + 489·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.34·7-s − 0.411·11-s − 0.426·13-s − 1.02·17-s − 0.0241·19-s − 1.03·23-s + 4/5·25-s + 0.192·29-s + 0.585·31-s − 1.81·35-s + 1.91·37-s + 0.114·41-s − 0.390·43-s + 1.02·47-s + 0.822·49-s + 1.60·53-s − 0.551·55-s − 1.45·59-s + 0.789·61-s − 0.572·65-s + 0.455·67-s + 0.601·71-s + 1.25·73-s + 0.555·77-s + 0.694·79-s + 0.646·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.867963828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.867963828\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 25 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 101 T + p^{3} T^{2} \) |
| 37 | \( 1 - 430 T + p^{3} T^{2} \) |
| 41 | \( 1 - 30 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 - 330 T + p^{3} T^{2} \) |
| 53 | \( 1 - 621 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 376 T + p^{3} T^{2} \) |
| 67 | \( 1 - 250 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 785 T + p^{3} T^{2} \) |
| 79 | \( 1 - 488 T + p^{3} T^{2} \) |
| 83 | \( 1 - 489 T + p^{3} T^{2} \) |
| 89 | \( 1 - 450 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1105 T + p^{3} T^{2} \) |
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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
−9.247204458725974793416278544461, −8.266769664436970503407941807189, −7.20486625028901444291783322961, −6.30795345537184846933462995507, −6.00843632866330852465018389541, −4.96563000226933331029469000817, −3.90561742116063638496522847829, −2.67881220184090618468696904049, −2.16069670954500208485082317390, −0.61855290661416154441323143999,
0.61855290661416154441323143999, 2.16069670954500208485082317390, 2.67881220184090618468696904049, 3.90561742116063638496522847829, 4.96563000226933331029469000817, 6.00843632866330852465018389541, 6.30795345537184846933462995507, 7.20486625028901444291783322961, 8.266769664436970503407941807189, 9.247204458725974793416278544461
