| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s + 6·13-s + 4·14-s + 16-s − 18-s + 4·21-s − 23-s + 24-s − 5·25-s − 6·26-s − 27-s − 4·28-s − 29-s − 8·31-s − 32-s + 36-s + 2·37-s − 6·39-s + 10·41-s − 4·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s + 0.872·21-s − 0.208·23-s + 0.204·24-s − 25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s − 0.185·29-s − 1.43·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s − 0.960·39-s + 1.56·41-s − 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.086346232514267307291222407424, −7.33574135193017491274962696316, −6.48835756453833685046997427127, −6.09762129822235473501575135060, −5.44160286452899607032939687636, −3.95432643292848018101273197395, −3.52792839035249660276402233750, −2.34051701016048875038352515307, −1.12380141780598365025220062322, 0,
1.12380141780598365025220062322, 2.34051701016048875038352515307, 3.52792839035249660276402233750, 3.95432643292848018101273197395, 5.44160286452899607032939687636, 6.09762129822235473501575135060, 6.48835756453833685046997427127, 7.33574135193017491274962696316, 8.086346232514267307291222407424
