L(s) = 1 | − 2-s − 3-s + 4-s − 3.05·5-s + 6-s − 7-s − 8-s + 9-s + 3.05·10-s − 3.44·11-s − 12-s + 14-s + 3.05·15-s + 16-s − 2.82·17-s − 18-s + 1.72·19-s − 3.05·20-s + 21-s + 3.44·22-s + 3.06·23-s + 24-s + 4.33·25-s − 27-s − 28-s + 3.85·29-s − 3.05·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.36·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.966·10-s − 1.03·11-s − 0.288·12-s + 0.267·14-s + 0.789·15-s + 0.250·16-s − 0.686·17-s − 0.235·18-s + 0.395·19-s − 0.683·20-s + 0.218·21-s + 0.733·22-s + 0.638·23-s + 0.204·24-s + 0.867·25-s − 0.192·27-s − 0.188·28-s + 0.715·29-s − 0.557·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2029380093\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2029380093\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3.05T + 5T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 + 0.978T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 8.04T + 47T^{2} \) |
| 53 | \( 1 + 8.33T + 53T^{2} \) |
| 59 | \( 1 + 9.97T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 4.75T + 67T^{2} \) |
| 71 | \( 1 + 3.66T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 4.49T + 79T^{2} \) |
| 83 | \( 1 - 7.15T + 83T^{2} \) |
| 89 | \( 1 + 7.90T + 89T^{2} \) |
| 97 | \( 1 + 9.09T + 97T^{2} \) |
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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
−7.85169517496273193734274844796, −7.40730342582341272001958503666, −6.68295960974621653870027853658, −6.03578280840861395865816257374, −4.97166540231491688536234537259, −4.51226761856362433534885958245, −3.38484262732808039572279310911, −2.84314240986558454635334546040, −1.51542214894382290893288209028, −0.26504010999771336569276909564,
0.26504010999771336569276909564, 1.51542214894382290893288209028, 2.84314240986558454635334546040, 3.38484262732808039572279310911, 4.51226761856362433534885958245, 4.97166540231491688536234537259, 6.03578280840861395865816257374, 6.68295960974621653870027853658, 7.40730342582341272001958503666, 7.85169517496273193734274844796