| L(s) = 1 | − 0.645·2-s − 2.49·3-s − 1.58·4-s + 1.61·6-s − 3.74·7-s + 2.31·8-s + 3.23·9-s + 6.02·11-s + 3.95·12-s − 3.48·13-s + 2.41·14-s + 1.67·16-s − 2.41·17-s − 2.09·18-s + 4.29·19-s + 9.34·21-s − 3.89·22-s − 7.88·23-s − 5.77·24-s + 2.24·26-s − 0.598·27-s + 5.92·28-s − 10.1·29-s − 5.58·31-s − 5.70·32-s − 15.0·33-s + 1.55·34-s + ⋯ |
| L(s) = 1 | − 0.456·2-s − 1.44·3-s − 0.791·4-s + 0.657·6-s − 1.41·7-s + 0.817·8-s + 1.07·9-s + 1.81·11-s + 1.14·12-s − 0.967·13-s + 0.645·14-s + 0.418·16-s − 0.585·17-s − 0.492·18-s + 0.985·19-s + 2.03·21-s − 0.829·22-s − 1.64·23-s − 1.17·24-s + 0.441·26-s − 0.115·27-s + 1.11·28-s − 1.87·29-s − 1.00·31-s − 1.00·32-s − 2.62·33-s + 0.267·34-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 | \( 1 + 0.645T + 2T^{2} \) |
| 3 | \( 1 + 2.49T + 3T^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 + 7.88T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 2.37T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 - 1.44T + 43T^{2} \) |
| 47 | \( 1 - 7.27T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 6.26T + 67T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 + 5.42T + 83T^{2} \) |
| 89 | \( 1 - 7.49T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.38774629018373998275092118185, −7.08674200060690736923724399250, −6.14066823203291570017278004539, −5.77936183895789468520419874492, −4.93307837901643286942064317558, −3.98570292645442774988142080225, −3.64222076517692229620204970119, −2.00934482422303112755993461923, −0.797799801732137730925480950864, 0,
0.797799801732137730925480950864, 2.00934482422303112755993461923, 3.64222076517692229620204970119, 3.98570292645442774988142080225, 4.93307837901643286942064317558, 5.77936183895789468520419874492, 6.14066823203291570017278004539, 7.08674200060690736923724399250, 7.38774629018373998275092118185
