| L(s) = 1 | − 1.84·2-s − 3-s + 2.41·4-s − 0.765·5-s + 1.84·6-s − 2.61·8-s + 9-s + 1.41·10-s + 1.84·11-s − 2.41·12-s + 13-s + 0.765·15-s + 2.41·16-s − 1.84·18-s − 1.84·20-s − 3.41·22-s + 2.61·24-s − 0.414·25-s − 1.84·26-s − 27-s − 1.41·30-s − 1.84·32-s − 1.84·33-s + 2.41·36-s − 39-s + 1.99·40-s + 0.765·41-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 3-s + 2.41·4-s − 0.765·5-s + 1.84·6-s − 2.61·8-s + 9-s + 1.41·10-s + 1.84·11-s − 2.41·12-s + 13-s + 0.765·15-s + 2.41·16-s − 1.84·18-s − 1.84·20-s − 3.41·22-s + 2.61·24-s − 0.414·25-s − 1.84·26-s − 27-s − 1.41·30-s − 1.84·32-s − 1.84·33-s + 2.41·36-s − 39-s + 1.99·40-s + 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3563155427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3563155427\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.84T + T^{2} \) |
| 5 | \( 1 + 0.765T + T^{2} \) |
| 11 | \( 1 - 1.84T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.765T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 0.765T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 0.765T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 - 1.84T + T^{2} \) |
| 89 | \( 1 - 1.84T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.258194332718216769491651175623, −8.828962717897983420315390962350, −7.82709331139753745625786885912, −7.23519278534519867206368702435, −6.39177245055586919686402035675, −5.98866102471959185287077694309, −4.37088654501690033479462863519, −3.49757022111972678230944710368, −1.75262866980429723893497749408, −0.873912551074877896167749507463,
0.873912551074877896167749507463, 1.75262866980429723893497749408, 3.49757022111972678230944710368, 4.37088654501690033479462863519, 5.98866102471959185287077694309, 6.39177245055586919686402035675, 7.23519278534519867206368702435, 7.82709331139753745625786885912, 8.828962717897983420315390962350, 9.258194332718216769491651175623
