| L(s) = 1 | + 0.543·2-s + 1.35·3-s − 1.70·4-s − 3.27·5-s + 0.737·6-s + 0.540·7-s − 2.01·8-s − 1.15·9-s − 1.78·10-s − 5.84·11-s − 2.31·12-s + 13-s + 0.293·14-s − 4.45·15-s + 2.31·16-s − 3.80·17-s − 0.627·18-s + 7.74·19-s + 5.59·20-s + 0.733·21-s − 3.17·22-s − 3.38·23-s − 2.73·24-s + 5.75·25-s + 0.543·26-s − 5.64·27-s − 0.920·28-s + ⋯ |
| L(s) = 1 | + 0.384·2-s + 0.784·3-s − 0.852·4-s − 1.46·5-s + 0.301·6-s + 0.204·7-s − 0.711·8-s − 0.385·9-s − 0.563·10-s − 1.76·11-s − 0.668·12-s + 0.277·13-s + 0.0784·14-s − 1.14·15-s + 0.579·16-s − 0.923·17-s − 0.147·18-s + 1.77·19-s + 1.25·20-s + 0.160·21-s − 0.676·22-s − 0.706·23-s − 0.557·24-s + 1.15·25-s + 0.106·26-s − 1.08·27-s − 0.174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.543T + 2T^{2} \) |
| 3 | \( 1 - 1.35T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 - 0.540T + 7T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 9.00T + 43T^{2} \) |
| 47 | \( 1 - 3.30T + 47T^{2} \) |
| 53 | \( 1 - 1.34T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 - 5.80T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 4.05T + 83T^{2} \) |
| 89 | \( 1 - 4.02T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−11.00879187889783750353330616015, −9.722497887814035503178474947696, −8.778236950577429760554613052971, −7.945801792845688402434469637852, −7.60394113149097418775315644186, −5.69387845062319568632379909458, −4.72840820955800662281117708492, −3.68519762962755382560002282184, −2.83362390177874144330467321193, 0,
2.83362390177874144330467321193, 3.68519762962755382560002282184, 4.72840820955800662281117708492, 5.69387845062319568632379909458, 7.60394113149097418775315644186, 7.945801792845688402434469637852, 8.778236950577429760554613052971, 9.722497887814035503178474947696, 11.00879187889783750353330616015
