| L(s) = 1 | + 0.482·2-s + 0.143·3-s − 7.76·4-s − 10.0·5-s + 0.0691·6-s − 7.61·8-s − 26.9·9-s − 4.85·10-s + 35.9·11-s − 1.11·12-s − 47.8·13-s − 1.43·15-s + 58.4·16-s + 53.5·17-s − 13.0·18-s − 49.7·19-s + 78.0·20-s + 17.3·22-s − 161.·23-s − 1.08·24-s − 23.9·25-s − 23.1·26-s − 7.72·27-s − 225.·29-s − 0.694·30-s − 139.·31-s + 89.1·32-s + ⋯ |
| L(s) = 1 | + 0.170·2-s + 0.0275·3-s − 0.970·4-s − 0.898·5-s + 0.00470·6-s − 0.336·8-s − 0.999·9-s − 0.153·10-s + 0.986·11-s − 0.0267·12-s − 1.02·13-s − 0.0247·15-s + 0.913·16-s + 0.764·17-s − 0.170·18-s − 0.601·19-s + 0.872·20-s + 0.168·22-s − 1.46·23-s − 0.00926·24-s − 0.191·25-s − 0.174·26-s − 0.0550·27-s − 1.44·29-s − 0.00422·30-s − 0.807·31-s + 0.492·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1656780544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1656780544\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 0.482T + 8T^{2} \) |
| 3 | \( 1 - 0.143T + 27T^{2} \) |
| 5 | \( 1 + 10.0T + 125T^{2} \) |
| 11 | \( 1 - 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 212.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.9T + 7.95e4T^{2} \) |
| 53 | \( 1 + 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 885.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 38.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 117.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 102.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 431.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3T + 9.12e5T^{2} \) |
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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
−8.641121334113479188726466903964, −7.933850452983373079455865039764, −7.34547633793907787399677023338, −6.09824473925993757872963700631, −5.50698134432369634795050489710, −4.49097039403755807616316816039, −3.85156752826581464442393716815, −3.14770924115584769487155244415, −1.74780016940818191212038631275, −0.17370730455734880764087820102,
0.17370730455734880764087820102, 1.74780016940818191212038631275, 3.14770924115584769487155244415, 3.85156752826581464442393716815, 4.49097039403755807616316816039, 5.50698134432369634795050489710, 6.09824473925993757872963700631, 7.34547633793907787399677023338, 7.933850452983373079455865039764, 8.641121334113479188726466903964
