L(s) = 1 | + 2-s + 1.31·3-s + 4-s − 5-s + 1.31·6-s − 3.61·7-s + 8-s − 1.28·9-s − 10-s − 3.14·11-s + 1.31·12-s + 3.52·13-s − 3.61·14-s − 1.31·15-s + 16-s + 4.96·17-s − 1.28·18-s − 1.16·19-s − 20-s − 4.74·21-s − 3.14·22-s + 1.31·24-s + 25-s + 3.52·26-s − 5.61·27-s − 3.61·28-s + 5.04·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.757·3-s + 0.5·4-s − 0.447·5-s + 0.535·6-s − 1.36·7-s + 0.353·8-s − 0.426·9-s − 0.316·10-s − 0.949·11-s + 0.378·12-s + 0.976·13-s − 0.966·14-s − 0.338·15-s + 0.250·16-s + 1.20·17-s − 0.301·18-s − 0.266·19-s − 0.223·20-s − 1.03·21-s − 0.671·22-s + 0.267·24-s + 0.200·25-s + 0.690·26-s − 1.08·27-s − 0.683·28-s + 0.936·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.927076847\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.927076847\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 + 0.957T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 + 8.77T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 - 7.30T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 1.58T + 89T^{2} \) |
| 97 | \( 1 + 0.135T + 97T^{2} \) |
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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
−8.208924763858887158825554408544, −7.50600363231935068855511576577, −6.67373346122528744023650987829, −5.97815947568521407330083857435, −5.39882651677996374879552898270, −4.28814238984736546874957644076, −3.49723496689021407393758461341, −3.04745696333914890940790657226, −2.38126598697135760716393791233, −0.77721350664857510366045906094,
0.77721350664857510366045906094, 2.38126598697135760716393791233, 3.04745696333914890940790657226, 3.49723496689021407393758461341, 4.28814238984736546874957644076, 5.39882651677996374879552898270, 5.97815947568521407330083857435, 6.67373346122528744023650987829, 7.50600363231935068855511576577, 8.208924763858887158825554408544