| L(s) = 1 | + 1.56·3-s − 0.592·5-s + 4.71·7-s − 0.559·9-s + 2.89·11-s − 1.57·13-s − 0.926·15-s − 17-s + 1.65·19-s + 7.37·21-s − 3.29·23-s − 4.64·25-s − 5.56·27-s + 6.83·29-s + 7.77·31-s + 4.52·33-s − 2.79·35-s + 4.36·37-s − 2.45·39-s − 2.41·41-s + 6.16·43-s + 0.331·45-s + 8.40·47-s + 15.2·49-s − 1.56·51-s + 12.9·53-s − 1.71·55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 0.265·5-s + 1.78·7-s − 0.186·9-s + 0.872·11-s − 0.435·13-s − 0.239·15-s − 0.242·17-s + 0.380·19-s + 1.60·21-s − 0.686·23-s − 0.929·25-s − 1.07·27-s + 1.27·29-s + 1.39·31-s + 0.787·33-s − 0.472·35-s + 0.718·37-s − 0.392·39-s − 0.376·41-s + 0.940·43-s + 0.0494·45-s + 1.22·47-s + 2.18·49-s − 0.218·51-s + 1.77·53-s − 0.231·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.218653454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.218653454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 0.592T + 5T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 - 2.89T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 - 6.83T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 + 2.41T + 41T^{2} \) |
| 43 | \( 1 - 6.16T + 43T^{2} \) |
| 47 | \( 1 - 8.40T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 61 | \( 1 - 3.04T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 5.29T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 1.52T + 97T^{2} \) |
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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
−8.515014262246428480237437430892, −7.78308657597252750482568586895, −7.35763373696372020948580413794, −6.21112669255234171935684598741, −5.40584250943141808771150972221, −4.40536488392260649386412356688, −4.04578339506979343898607839674, −2.78660873499700894568424271657, −2.08992669397684544493511493111, −1.04975034143147933996067275776,
1.04975034143147933996067275776, 2.08992669397684544493511493111, 2.78660873499700894568424271657, 4.04578339506979343898607839674, 4.40536488392260649386412356688, 5.40584250943141808771150972221, 6.21112669255234171935684598741, 7.35763373696372020948580413794, 7.78308657597252750482568586895, 8.515014262246428480237437430892
