| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4.44·7-s − 8-s + 9-s + 3.44·11-s − 12-s + 2.44·13-s + 4.44·14-s + 16-s − 4.44·17-s − 18-s + 19-s + 4.44·21-s − 3.44·22-s + 23-s + 24-s − 2.44·26-s − 27-s − 4.44·28-s − 4.34·29-s − 3·31-s − 32-s − 3.44·33-s + 4.44·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.68·7-s − 0.353·8-s + 0.333·9-s + 1.04·11-s − 0.288·12-s + 0.679·13-s + 1.18·14-s + 0.250·16-s − 1.07·17-s − 0.235·18-s + 0.229·19-s + 0.970·21-s − 0.735·22-s + 0.208·23-s + 0.204·24-s − 0.480·26-s − 0.192·27-s − 0.840·28-s − 0.807·29-s − 0.538·31-s − 0.176·32-s − 0.600·33-s + 0.763·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + 0.898T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 - 7.79T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 - 2.10T + 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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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
−8.751451936220509576523778186765, −7.43433276418775594442952202263, −6.88607281996833567728995304571, −6.18672088483711867071126449190, −5.76122180477598959985503379021, −4.28122472671208880741835202545, −3.58456769113566184992346954775, −2.52987735606926522428771591853, −1.19149362302376323593201201087, 0,
1.19149362302376323593201201087, 2.52987735606926522428771591853, 3.58456769113566184992346954775, 4.28122472671208880741835202545, 5.76122180477598959985503379021, 6.18672088483711867071126449190, 6.88607281996833567728995304571, 7.43433276418775594442952202263, 8.751451936220509576523778186765
