| L(s) = 1 | − 0.736·2-s − 1.41·3-s − 1.45·4-s − 5-s + 1.03·6-s − 7-s + 2.54·8-s − 1.00·9-s + 0.736·10-s − 5.32·11-s + 2.05·12-s − 0.724·13-s + 0.736·14-s + 1.41·15-s + 1.04·16-s + 3.52·17-s + 0.742·18-s + 1.06·19-s + 1.45·20-s + 1.41·21-s + 3.91·22-s − 0.370·23-s − 3.59·24-s + 25-s + 0.533·26-s + 5.65·27-s + 1.45·28-s + ⋯ |
| L(s) = 1 | − 0.520·2-s − 0.814·3-s − 0.729·4-s − 0.447·5-s + 0.424·6-s − 0.377·7-s + 0.899·8-s − 0.336·9-s + 0.232·10-s − 1.60·11-s + 0.593·12-s − 0.200·13-s + 0.196·14-s + 0.364·15-s + 0.260·16-s + 0.855·17-s + 0.175·18-s + 0.244·19-s + 0.326·20-s + 0.307·21-s + 0.835·22-s − 0.0771·23-s − 0.733·24-s + 0.200·25-s + 0.104·26-s + 1.08·27-s + 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 0.736T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 + 0.724T + 13T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 + 0.370T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 9.99T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 - 0.547T + 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 - 0.0183T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 9.07T + 61T^{2} \) |
| 67 | \( 1 - 0.894T + 67T^{2} \) |
| 71 | \( 1 + 4.15T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 9.94T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.47455589128631197270293507394, −7.07476447582565164459670362917, −5.86462965719749589515072048977, −5.29094449054789569856580909077, −5.02746312011302233692911471672, −3.86277816364092800947829030575, −3.23193809417697212498077949597, −2.10782543833181625542159229347, −0.72214412929681704489635233481, 0,
0.72214412929681704489635233481, 2.10782543833181625542159229347, 3.23193809417697212498077949597, 3.86277816364092800947829030575, 5.02746312011302233692911471672, 5.29094449054789569856580909077, 5.86462965719749589515072048977, 7.07476447582565164459670362917, 7.47455589128631197270293507394
