L(s) = 1 | − 2-s + 3-s + 4-s − 3.83·5-s − 6-s − 3.29·7-s − 8-s + 9-s + 3.83·10-s − 1.08·11-s + 12-s + 13-s + 3.29·14-s − 3.83·15-s + 16-s − 5.28·17-s − 18-s + 7.52·19-s − 3.83·20-s − 3.29·21-s + 1.08·22-s − 4.86·23-s − 24-s + 9.69·25-s − 26-s + 27-s − 3.29·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.71·5-s − 0.408·6-s − 1.24·7-s − 0.353·8-s + 0.333·9-s + 1.21·10-s − 0.327·11-s + 0.288·12-s + 0.277·13-s + 0.881·14-s − 0.989·15-s + 0.250·16-s − 1.28·17-s − 0.235·18-s + 1.72·19-s − 0.857·20-s − 0.719·21-s + 0.231·22-s − 1.01·23-s − 0.204·24-s + 1.93·25-s − 0.196·26-s + 0.192·27-s − 0.623·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 3.83T + 5T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 - 6.22T + 79T^{2} \) |
| 83 | \( 1 - 2.49T + 83T^{2} \) |
| 89 | \( 1 - 5.79T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−7.62244123911284425671986330538, −7.14751360193525594299418396943, −6.35229092588951707371710619383, −5.52937279518425523203959805676, −4.22517435963387747353911448136, −3.87952746273639900958356060958, −3.04208494184992178749068768260, −2.43318935538383437149290873078, −0.915475189864899244638395204044, 0,
0.915475189864899244638395204044, 2.43318935538383437149290873078, 3.04208494184992178749068768260, 3.87952746273639900958356060958, 4.22517435963387747353911448136, 5.52937279518425523203959805676, 6.35229092588951707371710619383, 7.14751360193525594299418396943, 7.62244123911284425671986330538