| L(s) = 1 | + 2-s − 3-s + 4-s + 1.25·5-s − 6-s − 0.797·7-s + 8-s + 9-s + 1.25·10-s − 2.98·11-s − 12-s + 13-s − 0.797·14-s − 1.25·15-s + 16-s + 4.87·17-s + 18-s − 4.45·19-s + 1.25·20-s + 0.797·21-s − 2.98·22-s + 4.08·23-s − 24-s − 3.41·25-s + 26-s − 27-s − 0.797·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.563·5-s − 0.408·6-s − 0.301·7-s + 0.353·8-s + 0.333·9-s + 0.398·10-s − 0.899·11-s − 0.288·12-s + 0.277·13-s − 0.213·14-s − 0.325·15-s + 0.250·16-s + 1.18·17-s + 0.235·18-s − 1.02·19-s + 0.281·20-s + 0.174·21-s − 0.635·22-s + 0.852·23-s − 0.204·24-s − 0.682·25-s + 0.196·26-s − 0.192·27-s − 0.150·28-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 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 + 0.797T + 7T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 17 | \( 1 - 4.87T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 - 4.08T + 23T^{2} \) |
| 29 | \( 1 + 9.96T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 + 3.80T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 3.00T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 + 6.72T + 59T^{2} \) |
| 61 | \( 1 - 4.06T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 + 1.94T + 71T^{2} \) |
| 73 | \( 1 + 7.45T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.45857000658594901925419647062, −6.37677578906989114437044076108, −6.20531238772516294289378729319, −5.17238354652372775806071758813, −5.06876516070652567695615189457, −3.85589423925156487040166410414, −3.24892561459762713951666272009, −2.25805062126849823484771108438, −1.43577433646403294880035473240, 0,
1.43577433646403294880035473240, 2.25805062126849823484771108438, 3.24892561459762713951666272009, 3.85589423925156487040166410414, 5.06876516070652567695615189457, 5.17238354652372775806071758813, 6.20531238772516294289378729319, 6.37677578906989114437044076108, 7.45857000658594901925419647062
