| L(s) = 1 | + 3-s − 2·4-s + 7-s − 2·9-s + 3·11-s − 2·12-s − 5·13-s + 4·16-s + 2·19-s + 21-s − 6·23-s − 5·27-s − 2·28-s − 3·29-s + 4·31-s + 3·33-s + 4·36-s + 2·37-s − 5·39-s + 12·41-s + 10·43-s − 6·44-s − 9·47-s + 4·48-s + 49-s + 10·52-s − 12·53-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.38·13-s + 16-s + 0.458·19-s + 0.218·21-s − 1.25·23-s − 0.962·27-s − 0.377·28-s − 0.557·29-s + 0.718·31-s + 0.522·33-s + 2/3·36-s + 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s − 0.904·44-s − 1.31·47-s + 0.577·48-s + 1/7·49-s + 1.38·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.372345712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372345712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.47868447936830, −14.06632836323113, −13.78208225831638, −12.97964629810273, −12.39683059531805, −12.17679455089126, −11.33327456127316, −11.09417925496840, −10.08258979698607, −9.622376150256635, −9.420923504844392, −8.782498232981091, −8.260332462621407, −7.640713589006995, −7.488383244507293, −6.381181594641163, −5.881245304523935, −5.327415418868207, −4.521534512811851, −4.284267738823575, −3.504665902252482, −2.864957140429553, −2.198380750007634, −1.366135302753616, −0.4106712322231199,
0.4106712322231199, 1.366135302753616, 2.198380750007634, 2.864957140429553, 3.504665902252482, 4.284267738823575, 4.521534512811851, 5.327415418868207, 5.881245304523935, 6.381181594641163, 7.488383244507293, 7.640713589006995, 8.260332462621407, 8.782498232981091, 9.420923504844392, 9.622376150256635, 10.08258979698607, 11.09417925496840, 11.33327456127316, 12.17679455089126, 12.39683059531805, 12.97964629810273, 13.78208225831638, 14.06632836323113, 14.47868447936830
