| L(s) = 1 | − 4·7-s − 3·9-s + 2·11-s + 4·17-s + 2·19-s − 2·23-s + 6·29-s − 8·31-s + 6·37-s − 10·41-s + 4·43-s + 9·49-s − 6·53-s − 6·59-s + 2·61-s + 12·63-s − 4·67-s + 12·71-s − 2·73-s − 8·77-s + 8·79-s + 9·81-s − 12·83-s − 14·89-s + 10·97-s − 6·99-s + 101-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s + 0.603·11-s + 0.970·17-s + 0.458·19-s − 0.417·23-s + 1.11·29-s − 1.43·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 9/7·49-s − 0.824·53-s − 0.781·59-s + 0.256·61-s + 1.51·63-s − 0.488·67-s + 1.42·71-s − 0.234·73-s − 0.911·77-s + 0.900·79-s + 81-s − 1.31·83-s − 1.48·89-s + 1.01·97-s − 0.603·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120179910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120179910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.83463576356075, −14.51163681059376, −13.85187615491975, −13.56988649599236, −12.77158266000400, −12.33698516284611, −11.97776242602751, −11.28352397484942, −10.78001368909162, −9.999031752888894, −9.664292215367642, −9.193742984900002, −8.553188040514124, −7.993589113762939, −7.280102320788459, −6.688648760256179, −6.142201582459090, −5.714062910059549, −5.068744046260908, −4.158939540393202, −3.399917047563974, −3.162329646331160, −2.389931741952248, −1.350434650340167, −0.4097456852301207,
0.4097456852301207, 1.350434650340167, 2.389931741952248, 3.162329646331160, 3.399917047563974, 4.158939540393202, 5.068744046260908, 5.714062910059549, 6.142201582459090, 6.688648760256179, 7.280102320788459, 7.993589113762939, 8.553188040514124, 9.193742984900002, 9.664292215367642, 9.999031752888894, 10.78001368909162, 11.28352397484942, 11.97776242602751, 12.33698516284611, 12.77158266000400, 13.56988649599236, 13.85187615491975, 14.51163681059376, 14.83463576356075
