| L(s) = 1 | + 2.76·2-s + 1.64·3-s + 5.64·4-s + 4.53·6-s − 1.89·7-s + 10.0·8-s − 0.305·9-s + 9.27·12-s + 2.06·13-s − 5.24·14-s + 16.5·16-s + 3.62·17-s − 0.844·18-s + 1.28·19-s − 3.11·21-s − 3.36·23-s + 16.5·24-s + 5.71·26-s − 5.42·27-s − 10.7·28-s − 1.12·29-s + 4.60·31-s + 25.7·32-s + 10.0·34-s − 1.72·36-s + 1.80·37-s + 3.54·38-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 0.947·3-s + 2.82·4-s + 1.85·6-s − 0.717·7-s + 3.56·8-s − 0.101·9-s + 2.67·12-s + 0.572·13-s − 1.40·14-s + 4.14·16-s + 0.879·17-s − 0.199·18-s + 0.293·19-s − 0.679·21-s − 0.702·23-s + 3.37·24-s + 1.12·26-s − 1.04·27-s − 2.02·28-s − 0.208·29-s + 0.826·31-s + 4.54·32-s + 1.71·34-s − 0.287·36-s + 0.296·37-s + 0.574·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.839372322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.839372322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 - 1.64T + 3T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 - 1.80T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 + 0.0110T + 43T^{2} \) |
| 47 | \( 1 + 9.22T + 47T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 - 5.47T + 59T^{2} \) |
| 61 | \( 1 + 7.35T + 61T^{2} \) |
| 67 | \( 1 + 7.01T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 3.02T + 73T^{2} \) |
| 79 | \( 1 - 6.12T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.433773180137301999582525566206, −7.80022376533477804206103667506, −7.00247618352979737763471862697, −6.16902078142908824193481150575, −5.69060187869278214188792133812, −4.72241083268054230828940033542, −3.77666870298560996023995328266, −3.27529128293486376786057378951, −2.65150432684785943648411891789, −1.58946241737947470219751595996,
1.58946241737947470219751595996, 2.65150432684785943648411891789, 3.27529128293486376786057378951, 3.77666870298560996023995328266, 4.72241083268054230828940033542, 5.69060187869278214188792133812, 6.16902078142908824193481150575, 7.00247618352979737763471862697, 7.80022376533477804206103667506, 8.433773180137301999582525566206
