| L(s) = 1 | + 2-s − 0.414·3-s + 4-s − 5-s − 0.414·6-s + 8-s − 2.82·9-s − 10-s + 0.585·11-s − 0.414·12-s + 6.24·13-s + 0.414·15-s + 16-s + 2.17·17-s − 2.82·18-s − 2.82·19-s − 20-s + 0.585·22-s − 4·23-s − 0.414·24-s − 4·25-s + 6.24·26-s + 2.41·27-s + 3.82·29-s + 0.414·30-s + 8.41·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.239·3-s + 0.5·4-s − 0.447·5-s − 0.169·6-s + 0.353·8-s − 0.942·9-s − 0.316·10-s + 0.176·11-s − 0.119·12-s + 1.73·13-s + 0.106·15-s + 0.250·16-s + 0.526·17-s − 0.666·18-s − 0.648·19-s − 0.223·20-s + 0.124·22-s − 0.834·23-s − 0.0845·24-s − 0.800·25-s + 1.22·26-s + 0.464·27-s + 0.710·29-s + 0.0756·30-s + 1.51·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.502288064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502288064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 - 8.41T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 - 3.17T + 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.357242684863523601621490814172, −7.81482217846503692839619868585, −6.67160589632424802665183069765, −6.12960366232169309552587539628, −5.61481140138014339537119366685, −4.59284640421638352489922389822, −3.82854407500918605434377669613, −3.21095229930910939301805000483, −2.12116078321672226475304100173, −0.830714410529271811816801097912,
0.830714410529271811816801097912, 2.12116078321672226475304100173, 3.21095229930910939301805000483, 3.82854407500918605434377669613, 4.59284640421638352489922389822, 5.61481140138014339537119366685, 6.12960366232169309552587539628, 6.67160589632424802665183069765, 7.81482217846503692839619868585, 8.357242684863523601621490814172
