L(s) = 1 | + 2.30·2-s − 0.621·3-s + 3.29·4-s − 1.43·6-s − 0.210·7-s + 2.99·8-s − 2.61·9-s + 2.40·11-s − 2.05·12-s − 0.974·13-s − 0.485·14-s + 0.287·16-s + 0.931·17-s − 6.01·18-s − 4.45·19-s + 0.131·21-s + 5.53·22-s + 3.73·23-s − 1.85·24-s − 2.24·26-s + 3.48·27-s − 0.695·28-s − 6.80·29-s − 10.0·31-s − 5.32·32-s − 1.49·33-s + 2.14·34-s + ⋯ |
L(s) = 1 | + 1.62·2-s − 0.358·3-s + 1.64·4-s − 0.584·6-s − 0.0797·7-s + 1.05·8-s − 0.871·9-s + 0.724·11-s − 0.591·12-s − 0.270·13-s − 0.129·14-s + 0.0718·16-s + 0.225·17-s − 1.41·18-s − 1.02·19-s + 0.0286·21-s + 1.17·22-s + 0.777·23-s − 0.379·24-s − 0.440·26-s + 0.671·27-s − 0.131·28-s − 1.26·29-s − 1.80·31-s − 0.940·32-s − 0.259·33-s + 0.367·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 0.621T + 3T^{2} \) |
| 7 | \( 1 + 0.210T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 + 0.974T + 13T^{2} \) |
| 17 | \( 1 - 0.931T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 + 6.80T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 - 0.136T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 1.71T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 2.51T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 0.0938T + 89T^{2} \) |
| 97 | \( 1 - 5.72T + 97T^{2} \) |
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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
−7.26201888983122865890212104940, −6.80352211307725623102956928705, −6.06859777890692650243125079900, −5.43110144791522234888520117714, −5.00151435910342666033161983582, −3.96095302104661885070773438173, −3.53261364397017495689947038146, −2.59927757341469938194396631734, −1.72152946101527512765930715354, 0,
1.72152946101527512765930715354, 2.59927757341469938194396631734, 3.53261364397017495689947038146, 3.96095302104661885070773438173, 5.00151435910342666033161983582, 5.43110144791522234888520117714, 6.06859777890692650243125079900, 6.80352211307725623102956928705, 7.26201888983122865890212104940