| L(s) = 1 | + 2-s − 2.82·3-s + 4-s − 2.82·6-s + 8-s + 5.00·9-s − 11-s − 2.82·12-s − 4.24·13-s + 16-s + 2.82·17-s + 5.00·18-s + 4.24·19-s − 22-s + 6·23-s − 2.82·24-s − 5·25-s − 4.24·26-s − 5.65·27-s − 4·29-s + 7.07·31-s + 32-s + 2.82·33-s + 2.82·34-s + 5.00·36-s + 2·37-s + 4.24·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.63·3-s + 0.5·4-s − 1.15·6-s + 0.353·8-s + 1.66·9-s − 0.301·11-s − 0.816·12-s − 1.17·13-s + 0.250·16-s + 0.685·17-s + 1.17·18-s + 0.973·19-s − 0.213·22-s + 1.25·23-s − 0.577·24-s − 25-s − 0.832·26-s − 1.08·27-s − 0.742·29-s + 1.27·31-s + 0.176·32-s + 0.492·33-s + 0.485·34-s + 0.833·36-s + 0.328·37-s + 0.688·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.378613778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378613778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−10.11042783006774044806973272608, −9.416757680657536506691178342452, −7.76566383835216614641179076709, −7.18888155751611006972057629735, −6.27318266217535869909217754411, −5.38617944588913588926723289746, −5.04620414196187682688602921525, −3.95091820987185959332967484972, −2.56618247751545306602543247206, −0.890438696955777795245481394781,
0.890438696955777795245481394781, 2.56618247751545306602543247206, 3.95091820987185959332967484972, 5.04620414196187682688602921525, 5.38617944588913588926723289746, 6.27318266217535869909217754411, 7.18888155751611006972057629735, 7.76566383835216614641179076709, 9.416757680657536506691178342452, 10.11042783006774044806973272608
