| L(s) = 1 | + 2.61·2-s − 1.19·3-s + 4.82·4-s − 3.12·6-s − 3.61·7-s + 7.39·8-s − 1.56·9-s − 11-s − 5.77·12-s + 1.47·13-s − 9.45·14-s + 9.66·16-s + 3.27·17-s − 4.10·18-s + 19-s + 4.32·21-s − 2.61·22-s + 7.45·23-s − 8.84·24-s + 3.86·26-s + 5.46·27-s − 17.4·28-s + 1.02·29-s + 1.64·31-s + 10.4·32-s + 1.19·33-s + 8.54·34-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 0.690·3-s + 2.41·4-s − 1.27·6-s − 1.36·7-s + 2.61·8-s − 0.523·9-s − 0.301·11-s − 1.66·12-s + 0.410·13-s − 2.52·14-s + 2.41·16-s + 0.793·17-s − 0.966·18-s + 0.229·19-s + 0.944·21-s − 0.557·22-s + 1.55·23-s − 1.80·24-s + 0.757·26-s + 1.05·27-s − 3.30·28-s + 0.190·29-s + 0.296·31-s + 1.85·32-s + 0.208·33-s + 1.46·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.367611324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.367611324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 + 1.19T + 3T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 - 1.64T + 31T^{2} \) |
| 37 | \( 1 - 6.71T + 37T^{2} \) |
| 41 | \( 1 + 3.92T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 3.71T + 47T^{2} \) |
| 53 | \( 1 - 0.102T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 - 1.37T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.87206398156513052359705054723, −6.93982420015175822695656526621, −6.50954651883228446721407820079, −5.89130848720699191687686170029, −5.31440894633684197737887216306, −4.71526595171200632937052958880, −3.59488523642419846453576793796, −3.17991280945365025260895828545, −2.44501671634028124046580375152, −0.877322474571191553706240979994,
0.877322474571191553706240979994, 2.44501671634028124046580375152, 3.17991280945365025260895828545, 3.59488523642419846453576793796, 4.71526595171200632937052958880, 5.31440894633684197737887216306, 5.89130848720699191687686170029, 6.50954651883228446721407820079, 6.93982420015175822695656526621, 7.87206398156513052359705054723
