L(s) = 1 | + 2.90·3-s − 3.52·7-s + 5.42·9-s + 3.80·11-s + 2.62·13-s + 5.80·17-s − 5.05·19-s − 10.2·21-s − 0.474·23-s + 7.05·27-s − 2·29-s + 2.75·31-s + 11.0·33-s + 7.18·37-s + 7.61·39-s + 5.18·41-s − 1.95·43-s + 5.33·47-s + 5.42·49-s + 16.8·51-s + 5.37·53-s − 14.6·57-s + 5.05·59-s − 12.2·61-s − 19.1·63-s − 7.76·67-s − 1.37·69-s + ⋯ |
L(s) = 1 | + 1.67·3-s − 1.33·7-s + 1.80·9-s + 1.14·11-s + 0.727·13-s + 1.40·17-s − 1.15·19-s − 2.23·21-s − 0.0989·23-s + 1.35·27-s − 0.371·29-s + 0.494·31-s + 1.92·33-s + 1.18·37-s + 1.21·39-s + 0.809·41-s − 0.297·43-s + 0.777·47-s + 0.775·49-s + 2.36·51-s + 0.738·53-s − 1.94·57-s + 0.657·59-s − 1.56·61-s − 2.41·63-s − 0.948·67-s − 0.165·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.458038829\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.458038829\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 + 0.474T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 - 4.85T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.893714395259092785365948333817, −7.985965752575430432024303389683, −7.35757565867414757516501283713, −6.43118157873599571573687003388, −5.94878579842365968820477616613, −4.35854511843336284965666949344, −3.70162419160270926244400326940, −3.18477104255624701687650277508, −2.27229583425703584225485956923, −1.10224269720422118375847820065,
1.10224269720422118375847820065, 2.27229583425703584225485956923, 3.18477104255624701687650277508, 3.70162419160270926244400326940, 4.35854511843336284965666949344, 5.94878579842365968820477616613, 6.43118157873599571573687003388, 7.35757565867414757516501283713, 7.985965752575430432024303389683, 8.893714395259092785365948333817