| L(s) = 1 | + 2.80·2-s + 5.85·4-s + 3.04·7-s + 10.7·8-s + 2.93·11-s − 3.24·13-s + 8.54·14-s + 18.5·16-s − 2.15·17-s − 19-s + 8.23·22-s + 1.19·23-s − 9.09·26-s + 17.8·28-s + 1.77·29-s − 9.34·31-s + 30.3·32-s − 6.04·34-s + 1.15·37-s − 2.80·38-s − 8.57·41-s − 5.27·43-s + 17.1·44-s + 3.35·46-s + 2.35·47-s + 2.29·49-s − 18.9·52-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 2.92·4-s + 1.15·7-s + 3.81·8-s + 0.886·11-s − 0.900·13-s + 2.28·14-s + 4.63·16-s − 0.523·17-s − 0.229·19-s + 1.75·22-s + 0.249·23-s − 1.78·26-s + 3.37·28-s + 0.329·29-s − 1.67·31-s + 5.36·32-s − 1.03·34-s + 0.190·37-s − 0.454·38-s − 1.33·41-s − 0.804·43-s + 2.59·44-s + 0.494·46-s + 0.343·47-s + 0.327·49-s − 2.63·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.792581532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.792581532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 - 1.15T + 37T^{2} \) |
| 41 | \( 1 + 8.57T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 + 9.96T + 61T^{2} \) |
| 67 | \( 1 + 4.98T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 5.66T + 79T^{2} \) |
| 83 | \( 1 + 3.00T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 3.24T + 97T^{2} \) |
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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
−8.071056668055668515910690334523, −7.24455088512502562142293503430, −6.85193740281325428701340985297, −5.93872211806772234407242806608, −5.21411183120775026603201071002, −4.64937862756995684931921459804, −4.03892795128303882282703424575, −3.16934635859923543380146947119, −2.14916219650401031413050451141, −1.52816581099161486566518054274,
1.52816581099161486566518054274, 2.14916219650401031413050451141, 3.16934635859923543380146947119, 4.03892795128303882282703424575, 4.64937862756995684931921459804, 5.21411183120775026603201071002, 5.93872211806772234407242806608, 6.85193740281325428701340985297, 7.24455088512502562142293503430, 8.071056668055668515910690334523
