| L(s) = 1 | + 2-s − 1.45·3-s + 4-s + 0.791·5-s − 1.45·6-s − 2.33·7-s + 8-s − 0.877·9-s + 0.791·10-s − 11-s − 1.45·12-s − 0.354·13-s − 2.33·14-s − 1.15·15-s + 16-s − 7.65·17-s − 0.877·18-s + 5.56·19-s + 0.791·20-s + 3.40·21-s − 22-s − 2.85·23-s − 1.45·24-s − 4.37·25-s − 0.354·26-s + 5.64·27-s − 2.33·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.841·3-s + 0.5·4-s + 0.353·5-s − 0.594·6-s − 0.882·7-s + 0.353·8-s − 0.292·9-s + 0.250·10-s − 0.301·11-s − 0.420·12-s − 0.0982·13-s − 0.623·14-s − 0.297·15-s + 0.250·16-s − 1.85·17-s − 0.206·18-s + 1.27·19-s + 0.176·20-s + 0.742·21-s − 0.213·22-s − 0.595·23-s − 0.297·24-s − 0.874·25-s − 0.0694·26-s + 1.08·27-s − 0.441·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 - 0.791T + 5T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 13 | \( 1 + 0.354T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 5.56T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 7.22T + 29T^{2} \) |
| 31 | \( 1 + 8.21T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 + 3.25T + 41T^{2} \) |
| 47 | \( 1 - 7.03T + 47T^{2} \) |
| 53 | \( 1 + 1.79T + 53T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 - 6.17T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 1.01T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 2.90T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.698433253375682829715457295383, −9.010173373847913970958333753216, −7.64635348717832386825160023967, −6.78968251144269502981259327850, −5.94446194782446046894350137103, −5.48226492726630368890305500350, −4.36630229873557318741787636583, −3.25897534433778030287486055064, −2.09146265366661394365750407140, 0,
2.09146265366661394365750407140, 3.25897534433778030287486055064, 4.36630229873557318741787636583, 5.48226492726630368890305500350, 5.94446194782446046894350137103, 6.78968251144269502981259327850, 7.64635348717832386825160023967, 9.010173373847913970958333753216, 9.698433253375682829715457295383
