| L(s) = 1 | − 2-s + 3-s + 4-s + 0.705·5-s − 6-s − 8-s − 2·9-s − 0.705·10-s + 6.35·11-s + 12-s − 3.15·13-s + 0.705·15-s + 16-s + 6.10·17-s + 2·18-s − 3.36·19-s + 0.705·20-s − 6.35·22-s − 5.61·23-s − 24-s − 4.50·25-s + 3.15·26-s − 5·27-s − 7.44·29-s − 0.705·30-s − 8.66·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.315·5-s − 0.408·6-s − 0.353·8-s − 0.666·9-s − 0.223·10-s + 1.91·11-s + 0.288·12-s − 0.873·13-s + 0.182·15-s + 0.250·16-s + 1.48·17-s + 0.471·18-s − 0.772·19-s + 0.157·20-s − 1.35·22-s − 1.17·23-s − 0.204·24-s − 0.900·25-s + 0.617·26-s − 0.962·27-s − 1.38·29-s − 0.128·30-s − 1.55·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 0.705T + 5T^{2} \) |
| 11 | \( 1 - 6.35T + 11T^{2} \) |
| 13 | \( 1 + 3.15T + 13T^{2} \) |
| 17 | \( 1 - 6.10T + 17T^{2} \) |
| 19 | \( 1 + 3.36T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 + 7.44T + 29T^{2} \) |
| 31 | \( 1 + 8.66T + 31T^{2} \) |
| 37 | \( 1 + 9.71T + 37T^{2} \) |
| 43 | \( 1 + 1.74T + 43T^{2} \) |
| 47 | \( 1 - 0.460T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 - 0.805T + 61T^{2} \) |
| 67 | \( 1 - 9.05T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 1.63T + 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 - 3.34T + 97T^{2} \) |
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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
−8.126090407558875918419320560726, −7.53995231404695094286877082679, −6.73699972837471256861370556477, −5.94738280655977962152319790907, −5.32232842369411422445491655072, −3.83640498455620405756917937874, −3.50561091983473055290257891025, −2.15411604593436542828535355448, −1.62060778720559868307334122815, 0,
1.62060778720559868307334122815, 2.15411604593436542828535355448, 3.50561091983473055290257891025, 3.83640498455620405756917937874, 5.32232842369411422445491655072, 5.94738280655977962152319790907, 6.73699972837471256861370556477, 7.53995231404695094286877082679, 8.126090407558875918419320560726
