L(s) = 1 | + 2-s − 1.41·3-s + 4-s + 2.82·5-s − 1.41·6-s + 8-s − 0.999·9-s + 2.82·10-s − 2·11-s − 1.41·12-s − 4.00·15-s + 16-s − 1.41·17-s − 0.999·18-s − 7.07·19-s + 2.82·20-s − 2·22-s − 4·23-s − 1.41·24-s + 3.00·25-s + 5.65·27-s + 2·29-s − 4.00·30-s + 8.48·31-s + 32-s + 2.82·33-s − 1.41·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.816·3-s + 0.5·4-s + 1.26·5-s − 0.577·6-s + 0.353·8-s − 0.333·9-s + 0.894·10-s − 0.603·11-s − 0.408·12-s − 1.03·15-s + 0.250·16-s − 0.342·17-s − 0.235·18-s − 1.62·19-s + 0.632·20-s − 0.426·22-s − 0.834·23-s − 0.288·24-s + 0.600·25-s + 1.08·27-s + 0.371·29-s − 0.730·30-s + 1.52·31-s + 0.176·32-s + 0.492·33-s − 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.278450789\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.278450789\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−13.75897892764682158104944478298, −12.99181020642601878454218038210, −11.91815093620013056118481872743, −10.79739703594736369657619807538, −9.981854834006967455041537620725, −8.371218444153554891246935980326, −6.48255113005534387242997547576, −5.86451827160416506222883333068, −4.66415541315340900968485969565, −2.41717302449154006608420706146,
2.41717302449154006608420706146, 4.66415541315340900968485969565, 5.86451827160416506222883333068, 6.48255113005534387242997547576, 8.371218444153554891246935980326, 9.981854834006967455041537620725, 10.79739703594736369657619807538, 11.91815093620013056118481872743, 12.99181020642601878454218038210, 13.75897892764682158104944478298