| L(s) = 1 | + 3.23·3-s + 3.61·5-s − 3.61·7-s + 7.47·9-s + 2.23·11-s − 1.76·13-s + 11.7·15-s − 2·17-s − 2.38·19-s − 11.7·21-s + 7.85·23-s + 8.09·25-s + 14.4·27-s + 7.47·31-s + 7.23·33-s − 13.0·35-s + 2.47·37-s − 5.70·39-s + 5.47·41-s − 3.38·43-s + 27.0·45-s − 5.61·47-s + 6.09·49-s − 6.47·51-s − 6.94·53-s + 8.09·55-s − 7.70·57-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 1.61·5-s − 1.36·7-s + 2.49·9-s + 0.674·11-s − 0.489·13-s + 3.02·15-s − 0.485·17-s − 0.546·19-s − 2.55·21-s + 1.63·23-s + 1.61·25-s + 2.78·27-s + 1.34·31-s + 1.25·33-s − 2.21·35-s + 0.406·37-s − 0.914·39-s + 0.854·41-s − 0.515·43-s + 4.03·45-s − 0.819·47-s + 0.870·49-s − 0.906·51-s − 0.953·53-s + 1.09·55-s − 1.02·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.410175151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.410175151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 7.85T + 23T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 - 2.90T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 0.527T + 67T^{2} \) |
| 71 | \( 1 + 1.09T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.174219042400517228994150541491, −7.18052588150376319161151229214, −6.60558867386782790791415220250, −6.22140183808071607411485616430, −5.00240708287150799912263141461, −4.21831219214413574693486522106, −3.20878845391201600372438543250, −2.77243658066905564473602902394, −2.10896305221460408579469296416, −1.17311731675042142894800992252,
1.17311731675042142894800992252, 2.10896305221460408579469296416, 2.77243658066905564473602902394, 3.20878845391201600372438543250, 4.21831219214413574693486522106, 5.00240708287150799912263141461, 6.22140183808071607411485616430, 6.60558867386782790791415220250, 7.18052588150376319161151229214, 8.174219042400517228994150541491
